Factorials and divisibility

Number Theory Level 3

Consider the set of factorials 1 ! , 2 ! , 3 ! , … , 100 ! 1!, 2!, 3!, \ldots , 100! .

There are, clearly, 100 numbers in this set. How many of them are divisible by 2209?

Notation: ! ! is the factorial notation. For example, 8 ! = 1 × 2 × 3 × ⋯ × 8 8! = 1\times2\times3\times\cdots\times8 .


The answer is 7.

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Denton Young by Denton Young , 47, USA

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1 solution

Denton Young
Jan 3, 2017

2209 = 4 7 2 2209 = 47^2

So we need two multiples of 47 to be included in the factorial. The first one joins at 47!, the second at 94!

So the factorials from 94! through 100! satisfy the condition. There are 7 of those.

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