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

Please clearly explain all the concepts of lcm and hcf?

No more verification needed, its clear, if you look carefully comment of "@Vijay pandey".

I am partially understood. Can anyone explain clearly the basics?

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.

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.

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.

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.

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.

I think its answer given is wrong because highest number. Would be 23*14*13 = 4186.

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.