Explanation:

To find how many natural numbers give a remainder of 31 when 1186 is divided by them, follow these steps:

- The condition can be rewritten as \(1186 = n \cdot q + 31\), where \(n\) is the divisor.

- Rearrange to find \(n: n = \frac{1186 - 31}{q} \)

- Simplify to find \(n: n = \frac{1155}{q} \)

- \(n\) must be a divisor of 1155 that is greater than 31.

Now, find divisors of 1155:

- 1155 factors to \(3 \times 5 \times 7 \times 11\).

- Divisors of 1155 include: 1, 3, 5, 7, 11, 15, 21, 33, 35, 55, 77, 105, 165, 231, 385, and 1155.

Select divisors greater than 31: \(33, 35, 55, 77, 105, 165, 231, 385, 1155\).

- There are 9 divisors.

- Correct answer is Option: 4 - 9

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