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

Yes, you are right @Chiya.

Guys, Let's understand it by set theory. We have set A={a,b,c,d} & B={e,b,c,f,g}.and in this question abcd are factors of A and ebcfg are factors of B.

So hcf will be b.c and that's common in both factors.If we say lcm of A, B then It will contain b,c as common part of their lcm({b,c}, adefg) where ad belongs to A and efg belongs to B and we will call them uncommon or other factors of their lcm.and you know that how are they used to get a particular no.A or B. You can take ad as 13 and efg as 14.

So, lcm will be 23 .13.14 and hcf will be 23 and no.A is 23.13 and B is 23.14.

I think,

1st no can be 23 and 2nd will 23 * 13 * 14.
The conditions are satisfied.
But options are irrelevant.
So
2nd possibility
23*13 and 23*14.

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?

Perhaps this answer is not correct:

As we see HCF= 23.
LCM= 23*13*14.
Let the numbers be a and b.
HCF*LCM=a*b.
a*b=23*23*13*2*7.

So we have to take 23 as common in each number as it is HCF.
So a=23*m,
b=23*n,
now m*n=13*2*7.

As HCF is 23 so m and n should be co prime.
Hence the possible pairs of m and n = (13,14), (26,7), (91,2).
As the value of 91 will give the highest value so we have to select 91,2 as a solution.
m=91 n=2.
and a=2093 and b=46.

Verify yourself LCM of 2093 and 46 is 23*13*14 and HCF of it is 23.
So the biggest possible number is 2093.

In question factors of LCM is 23, 13, 14 (read question).

&in the both number 23 factor required for make HCF 23 so least number is 23*13& greater 23*14.

Frnds Try to solve it in reverse. You will understand what the question is.
Means find HCF and LCM of,

23*13=299 and 23*14=322.
Factors of 299 =13*(23).
Factors of 322=2*7*(23).
So HCF is 23 that is given in the question.

Next thing in question is other two factors of their LCM are 13 and 14. To understand this look at the factors of these two numbers (299 & 322).

13 is there in factors of 299 and 14 (2*7) is present in factors of 322.

This explanation will help you to understand what the question is saying. If you understood the question u can easily solve other questions like this.

Let first no. = 23a.
Second no. = 23b.
So lcm = 23 ab.
Now give attention to the question, that it is saying other to factors of lcm are 13 and 14 so it means there are only three factors in lcm which are 23,13 and 14 so I think that's why a=13 and b = 14.

Given hcf is 23.

Also given OTHER two factors of lcm are 13 and 14 (we know hcf is always a factor for lcm, other than that there are two factors 13 and 14). Hence 23 13 14 are the three factors of lcm ie lcm is the product of these three factors 23*13*14.

Also we know that product of 2 numbers =hcf*lcm.

So product=23* (23*13*14).

We can write it as 23*13*23*14. We know hcf is the highest number that commonly divides two numbers.

Here in this product we can seperate out the 2 numbers with the common highest factor 23 (as given in question).

They are 23*13 and 23*14.

This indicates these are the two different numbers whose highest common factor is 23.

From these largest one is 23*14.

Why the largest number can't be 23*13*14 and the smallest number is 23?