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

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.

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).

Please correct me, if I am wrong:

Let two numbers be A and B. It is given that their HCF is 23.

Therefore, A = 23 * x
and B = 23 * y, where x and y are coprime pairs, so according to the property of coprime LCM(x,y) = xy.

And we can find their LCM as follows: LCM(A, B) =LCM(23*x, 23*y) = 23 * x * y.

Therefore, x and y are the other two factors of LCM, and we are given that those 2 factors are 13 and 14 respectively.

Therefore x = 13 and y = 14.

And hence largest number = 23 * 14 = 322.

@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.

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.

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.

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.

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?

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.

Please correct me if I am wrong:

Let two numbers be A and B. It is given that their HCF is 23. Therefore, A = 23 * x
and B = 23 * y, where x and y are any integers.

And we can find their LCM as follows: LCM(A, B) = 23 * x * y.

Therefore, x and y are the other two factors of LCM, and we are given that those 2 factors are 13 and 14 respectively.

Therefore x = 13 and y = 14. And hence largest number = 23 * 14 = 322.