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

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).

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).

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?

Thanks @Chiya.

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.

I am not understanding it. Please help me to get it.

Yes, you are correct, Thanks @Dheeraj.

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.

Can anyone please make it clear with the basic concept, please?

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.