Multiple factorials, prime numbers and modular arithmetic

Number Theory Level 2

For which prime p p , where 3 < p < 20 3< p< 20 , does ( ( 3 ! ) ! ! ! ) ! ! ! ! ! ≡ 0 ( m o d p ) ((3!)!!!)!!!!! \equiv 0 \pmod{p} ?

Hint: Check out the multiple factorials rule .


The answer is 13.

Abd Fml<3 by Abd FmL<3 , 19, France

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1 solution

Abd Fml<3
Mar 5, 2019

( 3 ! ) = 3 × 2 = 6 (3!)=3\times 2=6

( 6 ! ! ! ) = 6 × 3 = 18 (6!!!)=6\times 3=18

( 18 ! ! ! ! ! ) = 18 × 13 × 8 × 3 = 5616 (18!!!!!)=18\times 13\times 8\times 3=5616

5616 ≡ 0 ( m o d 13 ) 5616\equiv 0 \pmod{13}

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