Question

If $n$ is a positive integer which is relatively prime to $6,$ and if $n$ has $8$ divisors, then $12n$ has

• A. 24 divisors
• B. 48 divisors
• C. 96 divisors
• D. 16 divisors

Answer 1

Answer: (B)

48 divisors

Answer 2

$n$ has $8$ divisors and $12$ has $(1, 2, 3, 4, 6, 12) = 6$ divisors
$n$ and $6$ are relatively prime $\Rightarrow gcd (n, 6) = 1$
$2$ is a factor of $6 \Rightarrow gcd (n, 2) = 1$
since $gcd (n, 6) = 1$ and $gcd (n, 2) = 1$
$\Rightarrow \, gcd ( n, 12) =1$
$\Rightarrow \, n$ and $12$ are relatively prime.
$\therefore$ Number of divisors of $12n =$ Number of divisors of 12 $\times$ Number of divisors of $n$
$= 6 \times 8 = 48$