Question

Find the total number of integers for n such that $2\le n\le 2000$ and H.C.F. of $n$ and $36is1.$

Answer 1

36=22X32$36=2^2\times 3^2$

If H.C.F. of integer nn and 36 is 1, the n should not be divisible by $2$ or $3.$
Let us first find the numbers that are divisible by $2$ or $3.$
The number of integers in the range $[2,2000]$ that are divisible by $2$ is $1000(2,4,6,....1998,2000)$.
The number of integers in the range $[2,2000]$ that are divisible by $3$ is $666(3,6,\mathrm{9...},1995,1998).$
The number of integers in the range $[2,2000]$ that are divisible by $6$ is $333(6,12,\mathrm{18...},1992,1998).$
Total number of integers divisible by $2$ or $3$ is $1000+666-33=1333.$
Thus, the total number of integers that are divisible by neither $2$ nor $3$ is $1999-1333=666$.