3. Find all triples of consecutive natural numbers not exceeding 100, the product of which is divisible by 1001.
Answer: $76,77,78$ and $77,78,79$.

## Solution:

Since $1001=7 \cdot 11 \cdot 13$, among this triplet of numbers, there must be a number divisible by 7, a number divisible by 11, and a number divisible by 13. At the same time, a number cannot be divisible by both 11 and 13, so the divisibility by these numbers must be possessed by two different numbers. Since these are three consecutive numbers, the difference between the multiple of 11 and the multiple of 13 is no more than 2 (that is, they are either adjacent or one apart). We will start from the numbers that are multiples of 13.

If one number is 13, then the nearest multiple of 11 is 11, but then there must be a multiple of 7 between them - which is not the case.

If one number is 26, then the nearest multiple of 11 is 22, the distance between them is more than 2.

If one number is 39, then the nearest multiple of 11 is 44, the distance between them is more than 2.

If one number is 52, then the nearest multiple of 11 is 55, the distance between them is more than 2.

If one number is 65, then the nearest multiple of 11 is 66, but then the distance from 66 to the nearest multiple of 7 is more than 2.

If one number is 78, then the nearest multiple of 11 is 77, but then two numbers that are multiples of 13, 11, and 7 are already standing next to each other, so the third one can be any of the neighbors (76 or 79).

If one number is 91, then the nearest multiple of 11 is 88, the distance between them is more than 2.

There are no other numbers not exceeding 100 and divisible by 13, so the enumeration is complete.

If one number is 26, then the nearest multiple of 11 is 22, the distance between them is more than $2. If$ one number is 26, then the nearest multiple of 11 is 22, the distance between them is more than 2.

## Criteria:

Award 3 points for a complete answer with verification.

The absence of the enumeration provided in the work (possibly expressed by the phrase "we will enumerate all cases and verify that...", but the enumeration is not provided) does not add points to the already awarded for the answer.