Choose three consecutive whole numbers and multiply them. Why is the result always divisible by 6?
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The numbers consecutivos
Three numbers in a row always hide two guarantees: one provides factor 2 and another provides factor 3. That is why your product inevitably falls in the same place.
Hints
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- Between three numbers consecutivos siempre there are at least a number even.
- Between three numbers consecutivos siempre there is a multiple of 3.
- If a product has a factor 2 and a factor 3, is divisible by 6.
Solution
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Answer: because between three consecutive integers there is always a multiple of 2 and a multiple of 3. In any group of three consecutive numbers, one of them is even. That brings a factor 2 to the product. Also, in any group of three consecutive numbers, one of them is a multiple of 3. That contributes a factor of 3 to the product. Therefore, the product has factors of 2 and 3. As: $$
2\cdot 3=6,
$$ The product is always divisible by 6.
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