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

Please explain the answer.

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.

See for any number there are 1 HCF and 1 LCM.
So remember this formula:

HCF * LCM= N1 + N2.

HCF given 23;
And LCM given 13&14.
1st take 23*13= N1*N2.
Then take 24*14= N1*N2.
HIGHEST IS THE ANSWER.

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.

Thanks for your explanation @Chiya.

Thanks for your explanation @Jatin Lalwani.

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.

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.

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.

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.