Aptitude - Problems on H.C.F and L.C.M - Discussion

Discussion :: Problems on H.C.F and L.C.M - General Questions (Q.No.2)

2. 

The H.C.F. of two numbers is 23 and the other two factors of their L.C.M. are 13 and 14. The larger of the two numbers is:

[A]. 276
[B]. 299
[C]. 322
[D]. 345

Answer: Option C

Explanation:

Clearly, the numbers are (23 x 13) and (23 x 14).

Larger number = (23 x 14) = 322.


Abhinav said: (Sep 25, 2010)  
What is the concept behind this solution?

Sinchana said: (Dec 31, 2010)  
I didn't understand the solution can anybody explain for me.

Archana said: (Jan 1, 2011)  
Could not understand the concept, please help.

Poornima said: (Jan 27, 2011)  
HCF- Higest Common Factor, so we already have one factor of the number we need to find out ok. N we are also given the least common factors ie, 13 n 14. So to get the numbers just multiply both the Least common numbers with their HCF so that you can find the actual numbers.

Riya said: (Feb 27, 2011)  
Not convinced.

Prathyusha said: (Mar 6, 2011)  
1]HCF of two numbers is the number which is common factor for both numbers given.here 23 is the common factor.

2]other than this common factor ,we also will have the product of uncommon factors for the two numbers (13 and 14 here).

3]the first number =23*13
and second number=23*14

4]greatest of two numbers is definitely 23*14

Correct me if i am wrong please!.....

Pramod said: (Mar 15, 2011)  
Now what are these uncommon factors?

Chiya said: (Apr 22, 2011)  
HCF*LCM=a*b
23*(13*14)=a*b
(23*13)*(23*14)=a*b
a=23*13
b=23*14

Therefore b = 23*14 is the greatest.

Ramesh said: (Apr 30, 2011)  
I think the two numbers can be 23 and 23x13x14 i e 4186. Is it possible?
Suppose we take two numbers 4 and 24. Their HCF is 4 and LCM is 24. Then the greatest number is 24 itself even though it is the LCM.Am I right?

Avinash said: (May 9, 2011)  
What pratyusha said is h.c.f of two numbers that means common factor of two numbers is 23.

Hope there are more than common factors..
then what about uncommon factors..they given lcm of two factors..is uncommon factors are lcm of factors?

I want clear explanation

Anoop said: (Jun 11, 2011)  
Let first number = 23 *a and second number = 23 * b where a and b are the products of the uncommon factors and they will be different numbers.

lcm=23*13*14
that means 23*a*__=23*b*__ = 23 * 13 * 14
a*__ = 13*14
b*__ = 13*14
a and b are two different numbers
so either a=14 & b=13
or a=13 and b = 14

so the numbers are 23*13 and 23 *14

Parvez said: (Jul 2, 2011)  
Ya I understand but I'm not fully satisfied to this method.

Please give me suggestion.

#@Dev# said: (Aug 6, 2011)  
13 and 14 are two factors doesn't mean that 13 belongs to one no. and 14 belongs to another one

It may be like 26 belongs to greater no and 7 belongs to lower one.

Nitish said: (Sep 4, 2011)  
Chiya has given the right answer.

Neha said: (Sep 15, 2011)  
@chiya. Why is 23 multiplied twice? Please explain?

Shakti said: (Sep 25, 2011)  
@Chiya thanks for the solution.....

Hi_All said: (Oct 25, 2011)  
@Chiya thanq :-*

Eshant said: (Nov 11, 2011)  
I still didn't understnad the concept behind ths solution. Is their any easy method to solve these types of stuffs?

Lalitha said: (Dec 21, 2011)  
What chiya said is totally wrong
(H.C.F)*(L.C.M)=a*b this is right but
Chiya said: (Fri, Apr 22, 2011 02:18:39 AM)

HCF*LCM=a*b
23*(13*14)=a*b
(23*13)*(23*14)=a*b
a=23*13
b=23*14

Therefore b = 23*14 is the greatest.

23*(13*14)=a*b
(23*13)*(23*14)=a*b This is wrong.

Answer is right but logic is wrong. Anybody explain it clearly ?

Vivek said: (Feb 10, 2012)  
I have an idea that HCF+LCM=a*b

Bharath K S said: (Feb 12, 2012)  
They have given hcm of two numbers which is same for both the numbers..
hcm=23,23.
lcm of the two numbers are 13 and 14
lcm=13,14.
from this we should find the two unknown numbers x&y.
x=(lcm of x)*(hcm of x) therefore x=13*23=299.
y=(lcm of y)*(hcm of y) therefore y=14*23=322.
now we hav found the two unknown numbers and the biggest of them is 322, which is the ans...

Vinay said: (Mar 12, 2012)  
Not convinced, please try to give some more practical sollutions, which is easy to understand.

Dheeraj said: (Apr 19, 2012)  
Let the two numbers be a,b.
HCF(a,b) is always a factor of LCM(a,b). The other two factors of LCM(a,b) are 13 and 14. Therefore the three factors of LCM(a,b) are 23, 13, 14.
LCM(a,b)=LCM(13,14,23)=13*14*23.
Now HCF(a,b)*LCM(a,b)=a*b
23*(23*13*14)=a*b, Therefore bigger number=23*14 and smaller number=23*13 (Since 23 is a common factor of both numbers).

Yunus said: (Jul 4, 2012)  
If the two Nos are A and B.

LCM factor of A=13 and that of B=14 (by doing LCM).

And HCF of A and B is 23.

Than the No is always HCF*A =23*13 / HCF*B=23 *14.

Hence B=322 is larger No.

Ishu said: (Jul 8, 2012)  
Why l.c.m is a multiple of uncommon factors? if we take l.c.m of 2,4,6 then its l.c.m is 2*2*3=12 but when we multiply the uncommon factors i.e 2*3=6.

Darknight said: (Aug 23, 2012)  
Let &b r 2 nos.
NOW HCF*LCM=a*b
13 &14 are two factors of lcm ok
now lcm=a*13 (by defination of lcm && in the question it is given that there r 2 factors)
sim. lcm=b*14
23*lcm=a*b
23*a*13=a*b
so clearly b=23*13
a=23*13
solved!!!!

Sugandh said: (Aug 25, 2012)  
@Chiya explains good.

Siraj said: (Aug 29, 2012)  
@Chiya a*(b*c) is not equal to (a*b)*(b*c).... :D

Siraj said: (Aug 29, 2012)  
@Neha you are right. , how can we multiply 23 twice, I think Darknight's explanation is good. ,

Tej said: (Sep 2, 2012)  
Product of l.c.m and h.cf is that numbers so in the same way hcf is 23 and l.cm 13 and 4 are given and ther the ans 23*14 is right

Sanjeev Goel said: (Sep 19, 2012)  
With this concepts children will not be able to understand as they have been taught in school the other way.

The best solution is,

Since the factors of the lcm of 2 nos are 13, 14 then lcm of the 2 nos will be (13x14) = 182.

This means the lcm of the 2 nos is 182 answer as said that the hcf of the 2 nos is 23.

We know that LCM x HCF = product of the 2 nos.

So 182 x 23 = the product.

So product = 4186.

As factors are 13 and 14.

The nos are 4186/13 = 322 and 4186/14 = 299.

So the bigger of the 2 is 322.

Rohit said: (Sep 30, 2012)  
See definitions of H. C. F and L. C. M.
HCF=the product of common factor with least index.
LCM=product of all the prime factor of each of the given num with greatest index.
A/c to given question.
23 is the HCF of TWO given num.
And other two factor 13, 14 of the LCM.
Note :-here 14 is not a prime num hence 14=2*7.
It is clear that 23 is that prime num which is come in the LCM of two number.
Hence total prime number is (2, 7, 13, and 23).
Hence one num is 13*23=299.
And 2nd is=2*7*23.
Answer is (2*7*23).

Mohit said: (Jan 21, 2013)  
23 is common in both.

13 and 14 are factors of lcm thus may be common or may occur separate in both number but if they were commen then they should be included into hcf and that is not so.

Therefore 13 and 14 comes separately in both number.

Thus 23*13 and 23*14.

Shashwat Verma said: (Feb 19, 2013)  
The other solution is the bigger number is 23x7x13 and smaller number is 23x2.

These numbers also will have hcf as 23 and lcm as 23x13x14.

Sumi said: (Mar 24, 2013)  
@poornima.

Well explained.

hcf*lcm = n1*n2 = N1(let n1*n2=N1 for lcm=13).

Case1:
23*13=N1;

similarly,
Case2:
23*14=N2;(n1*n2=N2 for lcm=14).

So N2>N1.

Striker said: (May 31, 2013)  
H.C.F. = 23 and L.C.M. are 13 and 14. So if HCF is 23 then two numbers can be 23x and 23y And as given LCM are 13 &14 we know HCF*LCM=x*y so from that 23*13 and 23*14 are two number and greater one is 23*14.

Sairam said: (Jul 3, 2013)  
H.C.F of two numbers is 23(which is a common factor for both).

So the two numbers will be 23X , 23Y.

L.C.M*H.C.F = 23X*23Y.

Therefore L.C.M will be 23*x*y.

But the other two factors given are 13, 14.

So x will be 13 and y will be 14.

The numbers are 23X, 23Y which are 23*13, 23*14.

The greatest number is 23*14 = 322.

Vijay Pandey said: (Aug 8, 2013)  
When we calculate hcf of two numbers we find factors which are common.

And for lcm we take common factor and the remaining factors of two.

Here it is given that the other factors of lcm are 13 and 14

So numbers are hcf*lcm part

=>23*13 and 23*14 so 23*14 is greater.

Example: suppose we have two numbers 12 and 15.
Factor of 12=2*2*3.
Factor 0f 15=3*5.
So hcf =common factor=3.
And lcm=2*2*3*5=60 .
lcm part of 12=2*2.
lcm part of 15=5.
If we multiply these lcm parts by hcf we get the corresponding numbers.

Gannu said: (Aug 9, 2013)  
"HCF*LCM=Product of two numbers".

The above expression is universal.

Pole Star said: (Dec 7, 2013)  
Please clearly explain all the concepts of lcm and hcf?

Nitin said: (Feb 1, 2014)  
No more verification needed, its clear, if you look carefully comment of "@Vijay pandey".

Saravanan said: (Feb 13, 2014)  
I am partially understood. Can anyone explain clearly the basics?

Arun Krishnan said: (Apr 11, 2014)  
HCF is highest factor present in both numbers. Eg: HCF of 8,12 is 4.

Now this HCF must be multiplied with another number to get the the reqd number i.e 4*2 and 4*3.

LCM is least common number that we can obtain by multiplying some other number with the numbers we have. LCM of 8,12 is 24. i.e 8*3 and 12*2. LCM will also have the HCF in it. So, LCM = HCF * other factors, there will only be 2 other factors.

Now LCM*HCF = Product of numbers i.e 4*24 = 8*12; (4*2)*(4*3) = 8*12.

Gopakumar V said: (Apr 13, 2014)  
The answer need not be 23x14. The larger number can be 23x13x14 and the smaller number 23. In this case also HCF will be 23 and LCM will be 23x13x14. Only statement is the factors of LCM are 23,13 & 14. I think we cannot conclude the answer unless one of the numbers is clearly stated in the question. Please correct if I am wrong.

Mugil said: (May 5, 2014)  
From my view,

Read the question carefully,already we got h.c.f for two numbers. So,there is no dealing with h.c.f and keep h.c.f aside and change question like this (for your understanding). Find the l.c.m between 23 and 14 and also 23 and 13 and find which has greatest factor?

g.c.f for 23*13 = 299.
g.c.f for 23*14 = 322.

Out of 299 and 322, 322 is the greatest value and hence it is the answer.

Ahmed said: (Jul 2, 2014)  
The answer is simple, main problem is lack of information.

Lets consider,

Let the numbers, which are to be founded out, be x and y. Let their coefficients be a and b.

x and y are greater than 23.
23a=x, 23b=y.

LCM of x any y will be a same number.
Their LCM can be like this:
14x=13y.

Then:
23a(14)=23b(13)
a could be 13 and b could be 14 to satisfy the equations.
23(13)(14)=23(14)(13)
23=23.

Thus:
23b=23(14)=322=y=larger number.

NOTE:

With so lee information provided, the answer could be something else:
Put the equation again,
23a(14) = 23b(13).

a could be 26 and b could be 26.
23(28)(14)=23(26)(13).
23(2)=23(2).
46=46.

23a=23(28)=644=larger number.

HCF of 644 is 23, has 14 as the factor of its LCM with any number.

The question can be done, therefore, in many different ways.

Dipesh Das said: (Jul 17, 2014)  
Suppose we have 105 and 30.

On division by prime factorisation method.
We get HCF as 15 (i.e 3x5) and LCM as (3x5x7x2).

Clearly HCF is also a factor of LCM i.e HCF and other two factors when multiplied gives LCM and HCF multiplied with factors of the LCM give the numbers itself.

So the numbers are 15x7, and 15x2.

i.e 105 nd 30.

In the given question HCF is 23 and factors of LCM are 13 and 14.
So the numbers are 23x13 and 23x14, since 14 is greater, 23x14 is the greater number.

Deepak said: (Aug 27, 2014)  
I think its answer given is wrong because highest number. Would be 23*14*13 = 4186.

Pavan said: (Sep 7, 2014)  
I would like to solve this problem by using factors concept.
Here given least common multiple so we can write this numbers as 13:14.

This ratio are least parts of this numbers.
And hcf is given as 23 means highest parts 23.

So this numbers we can write as 23*13, 23*14.

Akhil said: (Sep 19, 2014)  
Probably its an option based question. We have to come to conclusion based on the option.

Clearly,
HCF * LCM = a*b;
Where a and b are the nos.

Here HCF = 23.
And LCM = 23*13*14.

So a*b = HCF * LCM.
= 23*(23*13*14).

Now we could separate the product in two ways:

Method 1:
---------
a*b = (23*13) * (23*14).
= 299 * 322.
Hence a = 299, b =322.

Method 2:
---------
a*b = 23*23*13*14.
= 23*23*13*2*7.
= 23*23*26*7.
= (23*26)*(23*7).
Hence a = 26 and b = 7.

HCF and LCM in both the methods 1 and 2 are same.

So we have to select any one pair of assumptions based on the options given in the question.

Saranyavathi said: (Sep 25, 2014)  
Please explain the basics of hcf and lcm?

Sharanjeet Kaur said: (Nov 1, 2014)  
HCF is common factor while LCM is greatest among the given numbers.

Cibi said: (Nov 1, 2014)  
L.C.M of two Numbers x1 = 13 & x2 = 14.
H.C.F (highest common factor) of x1 & x2 is = 23.

Therefore x1 = 23 & x2 = 23.

Largest Number is,

L.C.M(x1)*H.C.F(x1) = 13*23 = 299.
L.C.M(x2)*H.C.F(x2) = 14*23 = 322.

So answer = 322.

Sallu said: (Nov 6, 2014)  
First we have to understand the meaning of factor.

i.e. If number a is divided another number b exactly we say that a is factor of b.

Hence as per this factors are 13 and 14. So to find largest number.
Let us consider x and y are numbers.

x/factor = HCF.
y/factor = HCF.

x/13 = 23, x = 13*23 x = 299.
y/14 = 23, y = 14*23 y = 322.

So largest number is 322. Simple and less time consumption.

Manu said: (Dec 7, 2014)  
We mean (Number = H.C.F*L.C.M) it's wrong or not please explain me.

Vara said: (Feb 1, 2015)  
We know that product of two numbers = H.C.F*L.C.M.

X*Y = H.C.F*L.C.M.

In the question they gave H.C.F of two numbers is 23. And the factors of their L.C.M are 13 and 14.

X*Y = H.C.F*L.C.M | check | L.C.M factors.

X*Y = 23*(13*14) |of two no's.

So |X = 23*13 = 299; |23|322, 299 ||14, 13.

Y = 23*14 = 322; ||.

So the greatest among the two no's is 322 |.

Viswanadha Ashok said: (Feb 26, 2015)  
The L.C.M Of 2 numbers are (23x13) (23x14).

23x13 = 299, 23x14 = 322, which is largest number = 322.

Sanjay said: (Mar 7, 2015)  
I just want to know how these 13 and 14 obtained?

Hafizuddin said: (Mar 31, 2015)  
Let 'a' and 'b' be the two numbers.

HCF (a, b) = 23.

LCM (a, b) = 13*a = 14*b (or) LCM = 14*a = 13*b.

Note : The (or) case is just to indicate that we could have taken the factors the other way round ---> since they are factors for LCM. Both cases would result in the same large number.

Theorem: The product of two numbers = a*b = LCM (a, b)*HCF (a, b).

Applying the above theorem here :

=> a*b = HCF (a, b)*LCM (a, b).

=> a*b = 23*(13*a) (or).

Therefore => b = 23*13.

Similarly, a = 23*14.

The largest of the 2 numbers = 23*14 = 322 is the answer.

Rajesh said: (May 8, 2015)  
Read the question clearly.

Given that the two factors is 13 and 14 this means 23 is also a factor of L.C.M.

So LCM is 23*13*14.

Now 23 is HCF so suppose two numbers are 23*x and 23*y.

By theorem; (23*x)(23*y) = (23)(23*13*14).

This gives x equal to either 13 or 14 and similarly y.

Santhosh said: (Jun 1, 2015)  
Very good explanations.

Prasanna Kartik said: (Jul 9, 2015)  
Hi Guys,

My explanation is lets consider two numbers X, Y.

23 is the HCF of two numbers which means X and Y are exactly divisible by 23 in other words 23 is common factor for both the numbers so we can write X=23 X some value same as Y=23 X some value.

Then second part of the question if you see other remaining factors of LCM are 13 and 14 which means there is no common factors.

23|X , Y
|------
|13, 14

So X=23 X 13 and Y=23 X 14.

Correct me if I am wrong.

Sunil Kumar said: (Oct 2, 2015)  
LCM*HCF = Value of that number.

So, HCF(LCM1) = n1.

HCF*LCM2 = n2.

So 23*13 = n1.

23*14 = n2.

N1<N2.

Darshan said: (Dec 24, 2015)  
HCF means highest common factor.

LCM means least common Multiple.

Both are different. In this question they have given HCF of two numbers are 23.

Let us consider x and y as two no's then HCF (x, y) = 23.

We don't know the value of x and y. Then LCM of the two nos have factors 13 and 14.

This means that LCM (x, y) = z.

The value z have factors 13 and 14 which will divide z without leaving a remainder.

The main question is finding the larger of the two no is? That means we have to find biggest value among x and y.

I have understood the question but I don't how to solve it. I need clear explanation please.

Saurabh said: (Jan 15, 2016)  
Answer which is explained above assumes that the extra factors 13 and 14 are included in two separate numbers. The question does not say anything of this sort.

So technically, the correct answer is n1 = 23 and n2 = 23 x 13 x 14 = 4186.

You can verify that HCF = 23 and LCM = 2 x 7 x 13 x 23 = (2 x 7) x 13 x 23, for which the given statement that "The other two factors included in the LCM are 13 and 14" is perfectly valid.

Vidyut said: (Mar 2, 2016)  
Please clear the concept easily.

Rasim said: (Mar 12, 2016)  
I'm little bit confuse in this concept. How we know to solve this ways. Please anybody is there to clear my doubt.

Sankar said: (Apr 19, 2016)  
Dear Friends,

Still I didn't understand the concept of LCM & HCF.

Please explain me in an easy way.

Rose said: (Apr 26, 2016)  
I am confused with the question and method of solution. Can anyone help me?

Sameera said: (May 11, 2016)  
We have the terms HCF, LCM, product of 2 numbers (numbers be a,b).

In general,
Number = Factor * Factor.

eg :
6= 2*3 ----> 2, 3 factors of number 6.
8= 2*4-----> 2, 4 factors of 8.

So,
a = Factor * Factor.
b = Factor * Factor.

Here the 2 factors are 13, 14.

Given that, the highest common factor is 23,
So, we got another factor 23.

We require 4 factors to get a & b. But we have only 3.
And clearly said 23 is highest common factor.

Where 23 is a common factor in both a & b.

a = 23 * 13.
b = 23 * 14.

Therefore, the larger number = (23 x 14) = 322.

Hope you all understand this, if any mistake correct me.

Rahul said: (May 22, 2016)  
There should be given in the question that one factor is 14 and 13 is the another factor number.

Because, If the no. are x and y,
We can take two conditions according to question;

Condition 1:

x = 23 * 14.
y = 23 * 13 .

Condition 2:

x = 23 * 1
y = 23 * 14 * 1.

Then the only lcm will be 23 * 14 * 13, and then only you can write in the question that 14 and 13 are other two factors.

According to this question, the answer can be;

1)
x = 23*14.
y = 23*13.
So in this 23 * 14 will be greater.

2)
x = 23.
y = 23 * 13 * 14.

So, 23 * 13 * 14 will be greater.

According to me, the question is not complete and to complete it read my first line.

Hope you understand my explanation, Thankyou.

Jatin Lalwani said: (Jul 27, 2016)  
Let us say, the two numbers are: 23 x a and 23 x b.

Why? because 23 is the highest common " factor " of the two numbers. If you still don't understand why Study the concept of factors properly.

Now we know that,

"H.C.F is one of the factors of L.C.M". (That's why in the question they have mentioned, " the 'other two' factors are 13 and 14. That means the three factors of LCM are 23, 14, 13)" .

Therefore LCM = 23 x 14 x 13.

Now we know that, " Product of two numbers = Product of their H.C.F. and L.C.M ".
Therefore, (23 x a) x (23 x b) = 23 x 23 x 14 x 13.
Therefore, a x b = 14 x 13.
Therefore, a x b = 2 x 7 x 13.

Now there are more than one values possible for a and b which satisfy the above equation. But, only if we use a = 14 and b = 13 for the two numbers (23xa and 23xb), we will get the HCF = 23. For all other values of a and b, we won't get 23 as the HCF. Try it!

Therefore, The two numbers are 23 x 14 and 23 x 13.

Hope it is clear now!

Akshay said: (Aug 9, 2016)  
I understood it perfectly.

Here 23 is HCF. So it will also be a factor of LCM. And other two factors are given as 13 and 14 so now lcm will be 13 * 14 * 23. But we want to find numbers. So 1st number will be 23 * 13 because 23 is HCF and 13 is given as the 2nd factor of LCM. And the second number will be 23 * 14.

I hope you guys will understand this.

Erica Ferrao said: (Sep 15, 2016)  
a * b = lcm * (hfc)^2.
a * b = lcm * hfc * hfc.
a * b = (14 * 13) * 23 * 23,
a * b = (14 * 23) * (13 * 23),
a = 14*23 and b = 13 * 23,
a > b,

=>a = 322.

Correct me if I am wrong.

Tanaya said: (Sep 16, 2016)  
HCF is 23.

LCM is 13 * 14 * 23 (as it is given 13 & 14 are OTHER FACTORS),

Both the numbers have 13 & 14 as their factors, Hence the greater number should be divisible by 14 & the smaller number should be divisible by 13.

The answer is 322 (divisible by 14) as 322 = 23 * 14.

Chinni said: (Nov 15, 2016)  
@Chiya.

Good explanation and easy to understand, thanks for your explanation.

Smitten Ediot said: (Nov 17, 2016)  
As per the given information of the question,
We know HCM is 23 and LCM is 23 * 13*14.
Hence No 1 = 23 * 13, No2 = 23 *14.
Hence, 23*14 is the largest Number.

C T said: (Nov 25, 2016)  
Let Nos: x, y.
HCF = 23.

Other two factors of their L.C.M are 13 and 14 => x13 = LCM, y14 = LCM.
We know HCF *LCM = Product of Nos.
23 * x13 = xy = 23 * y14.
Take 1st pair, 23 * x13 = xy which gives y = 23 * 13.
2nd pair xy = 23 * 14.
x = 23 * 14 = 322 Highest.

Murali Banerjee said: (Nov 29, 2016)  
It's very simple. Without any calculation, it can easily be solved. Look at numbers.

13 & 14, the last digit of them are 3 & 4. All numbers. Except 322 are not divisible by 14. Hence 322 is the required number.

Hemanth said: (Dec 20, 2016)  
I didn't understand this method.

Please, can anyone explain me in an understanding manner?

Narendra said: (Jan 3, 2017)  
Not getting this. Please can anybody explain me clearly?

Rajesh Reddy said: (Jan 21, 2017)  
Statement 1:- We know that if 2 numbers are co-primes, then their LCM is equal to their product.
Statement 2:- We also know that if 2 numbers are co-primes,then their HCF is 1.

In the given question, 13 and 14 are the factors of the LCM of 2 numbers. Therefore it is understood that the 2 numbers are the multiples of 13 and 14.

23 is the HCF of 2 numbers. Therefore 2 numbers are not co-primes.
Given that 23 is the common factor of the 2 numbers - 23x13 = 299 and 23 x 14 = 322.
LCM of 299 and 322 = 4186.13 and 14 are the factors of 4186.
HCF of 299 and 322 = 23.
Therefore, 23 x 14 = 322 is the highest number.

Rajesh Reddy said: (Jan 21, 2017)  
@Ramesh.

We know that all the factors of the number A are also the factors of all the multiples of the number A.
In the given problem, 13 and 14 are factors of the Least Common Multiple of 2 numbers, therefore 13 and 14 are also considered as factors of the two numbers.
The HCF of 2 numbers = 23.
Therefore the required numbers are 23 x 13 = 299 and 23 x 14 = 322.
Therefore 23x14=322 is the highest number.
HCF of 299 and 322 = 23,
LCM of 299 and 322 = 4186.
13 and 14 are the factors of 4186.

Palanivel said: (Feb 8, 2017)  
In my view, x and why are given two numbers.

Hcf of (x, y) =23;.
Lcm of (x, y) =13, 14;.

We know that;

Hcf x lcm=X x Y.
So, 23x13=299;.

23 x 14 =322 which is larger so 322 is the answer.

If any mistakes correct me.

Vicky Patil said: (Feb 12, 2017)  
Thanks @Vijay Pandey.

N Karthik said: (Mar 11, 2017)  
HCF * LCM
X * Y=. 23 * 13 =299.
X * Y= 23 * 14 =322 large.

Falak.. said: (Mar 30, 2017)  
Simple and easy method to understand the solution. Thanks @Sanjeev Goel.

Md Aquib Ansari Ho said: (May 4, 2017)  
Thanks @Sameera.

Suraj &Amp; Shani Chouhan said: (May 20, 2017)  
Thanks for the given solution @Chiya.

Shahriar said: (May 28, 2017)  
Let, A and B are the two numbers.

As 23 is the HCF of the two numbers.
We can assume that
a= 23*x, b=23*y, where x and y are the other factors of the numbers.
Now, LCM of a(23*x) and b(23*y) = 23*x*y.
It is given that the other factors of the l.c.m are 13 & 14.
So we can say that x= 13 and y = 14.

Therefore a = 23 * 13,
And b = 23 * 14.
So the larger number is 23*14 = 322.

Raj said: (Jun 3, 2017)  
Yes, you are right @Chiya.

Mukesh Bisht said: (Jun 7, 2017)  
Guys, Let's understand it by set theory. We have set A={a,b,c,d} & B={e,b,c,f,g}.and in this question abcd are factors of A and ebcfg are factors of B.

So hcf will be b.c and that's common in both factors.If we say lcm of A, B then It will contain b,c as common part of their lcm({b,c}, adefg) where ad belongs to A and efg belongs to B and we will call them uncommon or other factors of their lcm.and you know that how are they used to get a particular no.A or B. You can take ad as 13 and efg as 14.

So, lcm will be 23 .13.14 and hcf will be 23 and no.A is 23.13 and B is 23.14.

Mahendra said: (Jun 19, 2017)  
I think,

1st no can be 23 and 2nd will 23 * 13 * 14.
The conditions are satisfied.
But options are irrelevant.
So
2nd possibility
23*13 and 23*14.

Chandan Anand said: (Jun 29, 2017)  
No, @Chiya. Who said that 23*(13*14) = (23*13)*(23*14)?

Here 23*(13*14)=23*13*14. In side the bracket, there is "13*14" i.e. not (13+14). So we can no take that.
Now, coming to question, we take an example:
Suppose we have two numbers: 14,21.
HCF of 14 and 21= 7.
Now, LCM of (14 and 21) = 42= 2*3*7.
Here the three factors of the LCM(14,21)= 2,3,7.
So, according to the question, 2 and 3 are satisfying the condition, but 7 is not. Why is this?

Sajidur Rahman said: (Jul 14, 2017)  
Perhaps this answer is not correct:

As we see HCF= 23.
LCM= 23*13*14.
Let the numbers be a and b.
HCF*LCM=a*b.
a*b=23*23*13*2*7.

So we have to take 23 as common in each number as it is HCF.
So a=23*m,
b=23*n,
now m*n=13*2*7.

As HCF is 23 so m and n should be co prime.
Hence the possible pairs of m and n = (13,14), (26,7), (91,2).
As the value of 91 will give the highest value so we have to select 91,2 as a solution.
m=91 n=2.
and a=2093 and b=46.

Verify yourself LCM of 2093 and 46 is 23*13*14 and HCF of it is 23.
So the biggest possible number is 2093.

Abhi said: (Aug 14, 2017)  
In question factors of LCM is 23, 13, 14 (read question).

&in the both number 23 factor required for make HCF 23 so least number is 23*13& greater 23*14.

Ashutosh Tripathi said: (Sep 9, 2017)  
Frnds Try to solve it in reverse. You will understand what the question is.
Means find HCF and LCM of,

23*13=299 and 23*14=322.
Factors of 299 =13*(23).
Factors of 322=2*7*(23).
So HCF is 23 that is given in the question.

Next thing in question is other two factors of their LCM are 13 and 14. To understand this look at the factors of these two numbers (299 & 322).

13 is there in factors of 299 and 14 (2*7) is present in factors of 322.

This explanation will help you to understand what the question is saying. If you understood the question u can easily solve other questions like this.

Gaurav said: (Sep 14, 2017)  
Let first no. = 23a.
Second no. = 23b.
So lcm = 23 ab.
Now give attention to the question, that it is saying other to factors of lcm are 13 and 14 so it means there are only three factors in lcm which are 23,13 and 14 so I think that's why a=13 and b = 14.

Sreehari said: (Sep 18, 2017)  
Given hcf is 23.

Also given OTHER two factors of lcm are 13 and 14 (we know hcf is always a factor for lcm, other than that there are two factors 13 and 14). Hence 23 13 14 are the three factors of lcm ie lcm is the product of these three factors 23*13*14.

Also we know that product of 2 numbers =hcf*lcm.

So product=23* (23*13*14).

We can write it as 23*13*23*14. We know hcf is the highest number that commonly divides two numbers.

Here in this product we can seperate out the 2 numbers with the common highest factor 23 (as given in question).

They are 23*13 and 23*14.

This indicates these are the two different numbers whose highest common factor is 23.

From these largest one is 23*14.

Manish said: (Oct 4, 2017)  
Why the largest number can't be 23*13*14 and the smallest number is 23?

Priyanshi Pandey said: (Oct 6, 2017)  
I think it is 23*14 the largest one.

Thangamani said: (Oct 10, 2017)  
23*13 =299.

23*14=322 so this is the large.

Riku Pande said: (Oct 28, 2017)  
23*14=322 largest.

Ranbir Singh said: (Dec 9, 2017)  
We can represent any integer number in the form of: pq+r.
Where p is dividend, q is quotient, r is reminder.
So: A = pq1 + r;
B = pq2 + r;

HERE P=23 (HCF GIVEN).
LCM FACTOR GIVEN ,SO q1 =13,q2 =14 , REMAINDER (r=0).
So, A = 23*13 = 299 , B= 23*14= 322.
ANSWER = 322.

Shivu said: (Apr 19, 2018)  
Can anyone please make it clear with the basic concept, please?

Shahnawaz said: (Apr 26, 2018)  
Here, 'other two factors of their lcm should be taken into consideration carefully.

Suppose there are two numbers, 15 and 25.Its hcf is 5 and lcm is 75.now write its factors.. 3*5*5, among these factors one of three is hcf, so when we multiply by its hcf with remaining other two factors i.e 5*3 and 5*5 gives original number.

Chan said: (May 15, 2018)  
Yes, you are correct, Thanks @Dheeraj.

Suresh Mohanty said: (May 16, 2018)  
I am not understanding it. Please help me to get it.

Jayesh said: (May 24, 2018)  
Let's understand what lcm and HCF are;

HCF of 2 no is the greatest number can divide both;
so 36 = 2x18,
46=2x23.

Here HCF is 2 and remainings are related in the relation of coprime only as 18 and 23 are coprime factors of these both numbers.

Therefore;

number A = HCF x coprime a.
number B= HCFx coprime b(Formula 1).

Now move to question
we know that number will be HCF x co-prime number, therefore, numbers will be A=23 x a
B=23 x b(where a & b are co-primes)

Now look at the formula;

HCF x LCM= A x B(here A and B are the numbers).

We can write A= HCF x coprime a.
&
B=HCF x coprime b(by formula 1).

Put this in formula
HCF x LCM =(HCF x A)(HCF x B).

Now if we take hcf in right hand side.
LCM =HCFx A x HCFx B/HCF.
therefore LCM=HCF x ab (where a & b are co primes).

and we get formula here
LCM= HCF x coprimes ab (formula 2)

we have in question

LCM 's factor 13 &14 which are coprimes.
means lcm =13 x 14 x LCM (as formula 2 tells us).
therefore by starting formula;
HCF x a =A -> 23 x 13=A=299.

HCF x b=B -> 23 x 14=B=322.

Ashwini Poojary said: (Jun 10, 2018)  
Thanks @Chiya.

Deepak Mahato said: (Jun 25, 2018)  
Here we have 23 which is the HCF (highest common factor) of two no which is unknown to us. now here, also two more factors of the no`s are given 13 and 14.

Now it is easy to calculate those two numbers
23 and 13 are the factors of 23*13=299,
and 23 and 14 are the factors of 23*14=322,
now here HCF= 23 and the larger no=322 which is the answer... Am I right?

Himanshu Latwal said: (Jul 10, 2018)  
We have been given the LCM =23 * 13 * 14.
HCF=23.

So, the largest no possible no could be 23*13*14 (since numbers are 23 &23 * 13 * 14).

Raki said: (Jul 11, 2018)  
Suppose LCM.of 4,6 is 12, so here 12 is LCM...but other factors of 12 is (3,4),(6,2).
So the product of factors is 12. so In this problem indirectly LCM given with factors those are (13,14).

So, now use formula LCM * HCF= product of numbers.

LCM(13*14)*HCF(23)= a*b.
Here 182*23=a*b.
2366=a*b...
We can use any of two numbers in this 2366 but HCF 23..Here, we can only take product of 23, so( 23*13),(23*14).

Monalisha said: (Aug 4, 2018)  
let two numbers be 23x and 23y then
LCM=23xy.

Since other two factors are 13 and 14
LCM=23x13x14.
hence, the two no. are 23x13 and 23 x 14,
larger is =23 x 14=322.

Vinay Gudipati said: (Aug 8, 2018)  
Given that the HCF of the number is 23 and then given LCM factors of that number is 13 and 14 so we can multiple 23 with the given factors which is given highest value it is the large value and the solution for that problem.

23 * 13=299.
23 * 14=322.
So 322 is the solution according to the problem.

Maswood Khan said: (Aug 10, 2018)  
if A and B are the Numbers,
let x = HCF(A,B) then,
LCM(A,B) = x * A/x * B/x.

In question they have given the other two factor of LCM as 13 and 14, we don't need first factor because we already know it's 23(HCF of Number).
So we can say -
A = 23*13.
B = 23*14.

Pratyush said: (Aug 22, 2018)  
Here in the question it is given that, " Other two factors are 13 and 14".
It means L.C.M of the number will contain these two along with the given common factor 23.
Thus L.C.M can be:

L.C.M(a,b)=23*13*14;
Now, we know ;
L.C.M(a,b) * H.C.F(a,b)= a*b;
=> ( 23*13*14)*(23)=a*b; ----> 1
Now, as 23 is the common factor so both number must be a multiple of 23.
So, Let us say numbers are 23*x, and 23*y.
Now, a*b =23*x * 23*y ----> (ii)
Therefore, from (i) and (ii).
We can conclude, a=23*13 and that b=23* 14. Or, vice-versa.
And, obviously greater among them will be 23*14.

Zakir said: (Sep 3, 2018)  
Thanks for your explanation @Jatin Lalwani.

Slhubhnandan said: (Sep 12, 2018)  
Thanks for your explanation @Chiya.

Pranav Shandilya said: (Mar 25, 2019)  
HCF = 23, Factors of LCM = 13 & 14(You can also write 26 & 7)
LCM = 23*13*14 = 23*26*14.
Let a & b be two nos.

a*b = HCF * LCM.
= 23 * 23 * 13 * 14 =23 * 23 * 26 * 7.

Either you get (23*13 & 23*14) or (23*26 & 23*7).

In 1st case:
a = 23*13 = 299 & b = 23*14 = 322.

In 2nd case:
a = 23*26 = 498 & b = 23*7 = 166.

So Greater No is either 322 or 498.

Aumshesh Upparwar said: (May 3, 2019)  
See for any number there are 1 HCF and 1 LCM.
So remember this formula:

HCF * LCM= N1 + N2.

HCF given 23;
And LCM given 13&14.
1st take 23*13= N1*N2.
Then take 24*14= N1*N2.
HIGHEST IS THE ANSWER.

Sunny Kumar Sah said: (Jul 11, 2019)  
Dear all,

Very easy, Let`s see in a different way,

Here in this question, it is given that there are two numbers ( let them be A & B ) whose HCF is 23. It means that both the numbers have the highest common factor as 23. So, we can write A as 23 x m & B as 23 x n, where m & n are some other factors(not common) , Now, in the question it is also given that Other Factors of LCM of those numbers (i.e. Lcm of A & B) is 13 & 14,

So LCM of A & B will be AB,or the same can be written as LCM of (23 x m) & (23 x n) = (23 x m)(23 x n) , where other factors of LCM are m & n(except 23 because 23 is common in A & B) ,but given, that those other factors are 13 & 14. It means m=13,n=14. now it is very cleared that.

LCM of (23 x m) & (23 x n) is (23 x 13)(23 x 14), where larger number is (23 x 14) or 322.

Sunny Kumar Sah said: (Jul 11, 2019)  
Please explain the answer.

Anand said: (Aug 4, 2019)  
Here, it is given that HCF of the two numbers are 23.

Similarly factors of LCM are 13 & 14,
Le the numbers be a & b.
We know LCM = 13*a or LCM =14*b (vice versa).
We also know LCM * HCF = a*b.
Here, 1) LCM = 13*a & HCF =23.
So, (13*a) * 23= a*b.
implies, b=23*13!
2) LCM =14*b & HCF =23.
So, (14*b) * 23 =a*b.
Implies a =23*14!
Thus, the Highest number is a= 23*14=322.

Pratik said: (Dec 19, 2019)  
I will explain to you with Example.
Take two numbers 21 & 36.
21 - 7 *3 & 36 - 12 *3.
H.C.F. - 3.
They gave H.C.F. and give two factors of L.C.M. 7 and 12.
Okey now simply apply logic which is greater.

Hasnat Azim said: (Mar 9, 2020)  
We know,
LCM*HCF = N1*N2.

Given,
HCF = 23 & LCM= 23*13*14.

Now,
N1*N2= (23*X) * (23*Y).
23*23*13*14= 23*23*X*Y.
X*Y=14*13.

So, N1= 23*14= 322.

Ezhil said: (Apr 8, 2020)  
E. G.consider the 2 number are 10 & 15
HCF of these two number is 5.(which is given)
10/5=2 and 15/5=3.

Here 2 and 3 are the least common multiples of 10 and 15.

So, if we have to find the number we will multiply the HCF which is 5 with the Least common Multiples which are 2 and 3. So that we can get the numbers 10 and 15. In that, the highest number is 15..

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