The highest common factor of x and 15 = 3*5 is 3 --> x is a multiple of 3 but not a multiple of 5;
The highest common factor of x and 140 = 2^2*5*7 is 14 = 2*7 --> x is a multiple of 2 and 7 but not a multiple of 4 or 5.

From above x is a multiple of 2, 3, and 7 but not a multiple of 4 or 5.

660 = 2^2*3*5*11.
x = 2*3*7*someting.

x and 660 have in common 2 and 3, could have also 11 in common but not 4 or 5. Therefore, HCF(x, 660) = 6k, so HCF(x, 660) is 6 or 66.

Answer: E.
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let us evaluate
We know HCF is the least common factor of the two numbers

s.no Factors of X factor of given number HCF Indication/inference
1 X 15=3*5 3 X has 3 as a factor but not 5
2 X 140=2*2*5*7 2*7 X has 2&7 as a factor but not 4&5
3 X 660=2*2*3*5*11 ???

We know from s.no 1 and 2 X has factors 3,2 & 7 but not 5 &4 but we do not have any information about 11 , since we have been asked to find max possible HCF let us consider 11 as a factor of X

so max possible HCF in the given condition of x and 660 is 3*2*11=66 i.e answer E

15 = 3 * 5
x has 3 but not 5 as a factor.

140 = 2 * 2 * 5 * 7
x has 2 and 7 as factors but not 5 and not 2*2.

So x has 2, 3 and 7 as factors at least but not 5 and not 2*2.
660 = 2 * 2 * 3 * 5 * 11
So the two numbers certainly have 2 and 3 common so their HCF will be 6 at least.
They will not have 2*2 and 5 common since x does not have these as factors.
They could have 11 common too so their HCF will be 6*11 = 66 at the most.

Answer (E)
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1/ The highest common factor between X and 15 is 3. It means that X is multiple of 3 and not multiple of 5
2/ The highest common factor between X and 140 is 14. We know that 140 = 2^2*5*7. It means that X is multiple of 2 and 7, but not multiple of 2^2.
From 1 and 2 => X = 2*3*7*A (A is an integer)
We know that 660 = 2^2*3*5*11
The highest common factor between X and 660 should be the product of 2*3*B (B should be 1 or 11).

=> the possible highest common factor between X and 660 is 2*3*11 = 66

Answer E

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