Question

The total number of integers n such that $2\le n\le 2000$ and the H.C.F of n and 36 is equal to 1 is

• A. $665$
• B. $666$
• C. $667$
• D. None of these

Answer 1

The correct option is B $666$

We need to find integers from 2 to 2000 which are co-prime to 36 i.e. do not have 2 or 3 as factors.
Instead of counting from 2 to 2000, let us count from 1 to 1998 since 1998 is a multiple of both 2 and 3.
Thus, number of co-primes is given by $1998\ast (1-\frac{1}{2})\ast (1-\frac{1}{3})=1998\ast \frac{1}{2}\ast \frac{2}{3}=666$
Now, we need to remove 1 number (i.e. 1 itself) from this set of 666 numbers because it was not a part of the range given and we also need to consider one extra number 1999 which is co-prime to 36 (note that 2000, which we had not considered is not co-prime to 36, hence cannot be considered).
Thus, the answer is $666-1+1=666$
Hence, (B) is correct.