(C) Given that $d(n)$ represents the number of positive divisors of $n$.For a prime number $P$, the number of divisors of $P^7$ is $7+1 = 8$.So, $d(P^

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(C) Given that $d(n)$ represents the number of positive divisors of $n$.For a prime number $P$, the number of divisors of $P^7$ is $7+1 = 8$.So, $d(P^7) = 8$.Next, we find $d(d(P^7)) = d(8)$. Since $8 = 2^3$, the number of divisors is $3+1 = 4$.So, $d(d(P^7)) = 4$.Finally, we find $d(d(d(P^7))) = d(4)$. Since $4 = 2^2$, the number of divisors is $2+1 = 3$.Therefore, $d(d(d(P^7))) = 3$.

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For any integer $n \geq 1$, the number of positive divisors of $n$ is denoted by $d(n)$. Then, for a prime $P$, $d(d(d(P^7)))$ is equal to

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