Question

Find the total number of integer $n$ such that $2\le n\le 2000$ and H.C.F. of n and $36$ is $1$

Answer 1

$36=2^2{3}^2$

Since, $HCF(36,n)=1$

Therefore, $n$ should not be a multiple of $2$ or $3$

$2\le n\le 1000$

Total number $T=999$

Number of numbers divisible by $2=N_2$

Number of numbers divisible by $3=N_3$

Number of numbers divisible by $6=N_6$

Therefore, there are $T-N_2-N_3+N_6$ total integers

$a=2$

$l=1000$

$d=2$

We know,

$l=a+(N_2-1)d$

$N_2=\frac{1000-2}{2}+1=500$

Similarly,

$N_3=\frac{999-3}{2}+1=333$

$N_6=\frac{996-6}{6}+1=166$

Therefore,

$T-N_2-N_3+N_6=999-500-333+166=332$

So, the total number of integers are 332.