Factorials in a Set

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How many factorials of non-negative integers are in the set { 1 , 8 , 15 , 22 , 29 , 7 n + 1 , . . . } \left\{ {1, 8, 15, 22, 29, 7n + 1, ...}\right\} ?


The answer is 3.

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1 solution

Tristan Shin
Jan 23, 2014

Since the answer is 1 greater than a multiple of 7, it cannot be divisible by 7. Therefore, we only need to check all of them up to 6.

0 ! 1 m o d 7 0!\equiv 1 \bmod{7}

1 ! 1 m o d 7 1!\equiv 1 \bmod{7}

2 ! 2 1 m o d 7 2 m o d 7 2!\equiv 2*1 \bmod{7} \equiv 2 \bmod{7}

3 ! 3 2 m o d 7 6 m o d 7 3!\equiv 3*2 \bmod{7} \equiv 6 \bmod{7}

4 ! 4 6 m o d 7 3 m o d 7 4!\equiv 4*6 \bmod{7} \equiv 3 \bmod{7}

5 ! 5 3 m o d 7 1 m o d 7 5!\equiv 5*3 \bmod{7} \equiv 1 \bmod{7}

6 ! 6 1 m o d 7 6 m o d 7 6!\equiv 6*1 \bmod{7} \equiv 6 \bmod{7}

Therefore, the only ones that work are 0, 1, and 5. So the answer is 3 \boxed{3} .

Note to some people who have said that the answer is 2: 0 ! 0! is valid, as the question only said "non-negative" integers, meaning that 0 is allowed.

Tristan Shin - 7 years, 4 months ago

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