Question

# The number of proper divisors of . . 15^{r} is-

- (p + q + 1) (q + r + 1) (r + 1)
- (p + q + 1) (q + r + 1) (r + 1) – 2
- (p + q) (q + r) r – 2
- None of these

Hint:

### A proper divisor of a natural number is **the divisor that is strictly less than the number**.

For example, number 20 has 5 proper divisors: 1, 2, 4, 5, 10, 20

Proper divisors of number 20 are 2,4,5 and 10 excluding 1 and 20(the number itself)

## The correct answer is: (p + q + 1) (q + r + 1) (r + 1) – 2

### . . - We need to find proper divisors.

Suppose is a number then factors of = ( and a is proper

i.e. has total division = (n + 1)

Now, =

We know that

Thus, = =

Total factors = (p+q+1)(q+r+1)(r+1)

However, proper divisors exclude $1$ and the number itself.

Hence, the answer is $(p+q+1)(q+r+1)(r+1)−2.$

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