What is the remainder when (67!) is divided by 71 ?
Number Theory
Level
2
Find the remainder when (67!) divided by 71.
The answer is 12.
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1 solution
By Wilson’s theorem
70! ≡ −1 mod 71
We know that 71 is a prime number.
Since 70! = 70×69×68×67! , we have
70×69×68×67! ≡ −1 mod 71
or
(−1)×(−2)×(−3)×67! ≡−1 mod 71
or
6×67! ≡ 1 mod 71.
Multiplying both sides by 12
72×67! ≡ 12 mod 71
This means that
1×67! ≡ 12 mod 71.
or
67! ≡ 12 mod 71.
The answer is 12.