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

@All.

This question has a very clear concept of the fundamental theorem of arithmetic
Let one no. be =23a and the second no. =23b.

According to question;

First num. * Second num = hcf * lcm.
23a * 23b = 23 * lcm.

(Note the concept that HCF is always a factor of lcm )
So we can write;

23a* 23b= 23* 23* 13 * 14 ( as given in the question that 13 and 14 are the other factors )
so, ab = 13*14.

Now ACCORDING TO the FUNDAMENTAL THEOREM OF ARITHMETIC;

We need to express ab as the product of unique prime factors
ab = 13*7*2
Now, there can be 3 possible conditions
a can be
1. 13*7*2 ( then b will be 1) { note= you can do the same thing with b also there will be no change, only the last answer will be a=322}
2. 13*7 (then b will be 2)
3. 13 ( then b will be 14).

According to all the 3 conditions, the two numbers can be-
FOR 1nd condition.
first no. - 4186 ( 23a= 23 *13*7*2)
second no. - 23 (23b=23*1).

FOR 2nd condition.
First no. - 2743 (23a=23*13*7)
Second no. - 46 (23b = 23*2).

FOR 3 condition
First no. - 299 (23a = 23*13)
Second no. - 322 (23b=23*14)

So, if we see in the given question the options are given according to 3 condition
Hence, according to the 3rd condition, the greater number is b=322.
I hope it helps.

Given: HCF of A and B is 23.

Conclusion: Since 23 is the greatest common factor, both A and B must be multiples of 23.

Let A = 23 * x, where x is some integer.
Let B = 23 * y, where y is some integer.

Note: x and y don't have any common factors other than 1 (otherwise, the HCF would be greater than 23).

Finding the LCM:

LCM is the smallest number that is a multiple of both A and B.

A is a multiple of 23 and x.
B is a multiple of 23 and y.
To include all factors of both A and B, we need to include 23, x, and y in the LCM. Since x and y don't share any common factors, we simply multiply them all together.

Therefore, LCM(A, B) = 23 * x * y.
If x and y had a common factor other than 1, it would mean that the HCF of A and B would be greater than 23.
Here's why:
Let's say x and y share a common factor of 'd'.
So, x = d * p and y = d * q, where p and q don't have any common factors.
Now, A = 23 * x = 23 * d * p and B = 23 * y = 23 * d * q.
In this case, the HCF of A and B would be 23 * d, which is greater than 23.
This contradicts our initial assumption that the HCF is 23.
Therefore, for the formula LCM(A, B) = 23 * x * y to hold, x and y must not have any common factors other than 1.

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.

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!

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.

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.

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.

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.

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.

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.