The number of natural numbers n such that (n + 1)2/(n + 7) is an integer is:

Concept

To count integer values of a rational expression like (n + 1)

2/(n + 7), rewrite it by polynomial division as Quotient + Remainder/(n + 7). The expression is an integer exactly for those natural numbers n for which (n + 7) divides that fixed remainder -- i.e., for which (n + 7) is a divisor of the remainder.

Application

1. Expand (n + 1)2 = n2 + 2n + 1.

2. Divide by (n + 7): n2 + 2n + 1 = (n + 7)(n - 5) + 36, so the expression equals (n - 5) + 36/(n + 7).

3. For the expression to be an integer, (n + 7) must be a divisor of 36.

4. The positive divisors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36.

5. Since n is a natural number (n >= 1), (n + 7) >= 8, so only divisors of 36 that are at least 8 qualify: 9, 12, 18, 36.

6. Each qualifying divisor gives one valid n: n+7=9 -> n=2; n+7=12 -> n=5; n+7=18 -> n=11; n+7=36 -> n=29.

7. That is exactly 4 natural numbers.

Cross-check

n = 2: (3)2/9 = 9/9 = 1, an integer. n = 29: (30)2/36 = 900/36 = 25, an integer. Both confirm the count.

Result

There are 4 such natural numbers.