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

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?

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).

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).

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.

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.

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.

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.

Thanks for your explanation @Jatin Lalwani.

Thanks for your explanation @Chiya.

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.