Question

For a positive integer $n$, define $d(n) =$the number of positive divisors of $n$. What is the value of $d(d(d(12)))$?
A) $1$
B) $2$
C) $4$
D) None of these

Answer 1

First, evaluate the number of positive divisors of $12$ and that will be the value of $d(12)$. Similarly solve further and rewrite the expression after evaluating each value then again evaluate the number of positive divisors of the obtained numbers and continue this process until we get the value of $d(d(d(12)))$ .

Complete step-by-step answer:
We are given that for a positive integer $n$ and $d(n)$is defined as $d(n) =$the number of positive divisors of $n$.
We have to evaluate the value of $d(d(d(12)))$.
Since, in the innermost bracket $d(12)$ is present therefore, we start with $d(12)$.
$d(12) =$the number of positive divisors of $12$.
The positive divisors of $12$ are $1,2,3,4,6,12$, the total number of positive divisors of $12$ are $6$.
Therefore, $d(12) = 6$
Now, the expression becomes $d(d(12)) = d(d(6))$
Now, we evaluate $d(6)$
$d(6) =$the number of positive divisors of $6$.
The positive divisors of $6$ are $1,2,3,6$, the total number of positive divisors of $6$ are $4$.
Therefore, $d(6) = 4$
Now, the expression becomes $d(d(d(12))) = d(4)$
Now, we evaluate $d(4)$
$d(4) =$the number of positive divisors of $4$.
The positive divisors of $4$ are $1,2,4$, the total number of positive divisors of $4$ are $3$.
Therefore, $d(4) = 3$
Hence, $d(d(d(12))) = 3$
It does not match with any of the options.
Therefore, option (D) is correct.

Note: The divisors of any number are also called the factors of that number. The definition of the divisor is divisors are those numbers which divides the number completely with no remainder left.