24 24 6

Is it true that among any 24 consecutive integers, all larger than 5, there always exists a number which has at least three (not necessarily distinct) prime factors?

As an example, 12 has three prime factors: 2, 2, 3.

true false

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1 solution

Zee Ell
Nov 24, 2017

We will show, that a much stronger statement is true:

"Amongst any 8 positive consecutive integers, there is at least one, which has at least three (not necessarily distinct) prime factors."

It is easy to see, that amongst 8 consecutive integers, there is exactly one, which is divisible by 8. Since 8 (= 2×2×2) itself has 3 prime factors, therefore this number has at least 3 prime factors.

Hence, our answer should be:

TRUE \boxed { \text { TRUE } }

Nice. And this also works for the case of 3 distinct prime factors, where we will need 2 × 3 × 5 = 30 2 \times 3 \times 5 = 30 consecutive numbers.

Calvin Lin Staff - 3 years, 6 months ago

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