What is the remainder when 2 1 1 ! is divided by 2 1 3 ?
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211/71=2.972
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It's 211!. The ! symbol means factorial. The factorial s defined as the product of all positive integers less than or equal to the number. e.g.
4 ! = 4 × 3 × 2 × 1 = 2 4
Notice that 213 is equal to 3 times 71, which are both factors of 211!. That would mean, that 211! is a multiple of 213, so he remainder is 0.
211/71=2.972
By Wilson's Theorem ( ( n − 1 ) ! ≡ − 1 m o d ( n ) ) we have:
2 1 2 ! ≡ 2 1 2 m o d ( 2 1 3 )
Then 2 1 2 ! / 2 1 2 ≡ 2 1 2 / 2 1 2 m o d ( 2 1 3 )
2 1 1 ! ≡ 0 m o d ( 2 1 3 )
213=3*71,and these two factors are there in 211!,
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213=3*71,and these two factors are covered in 211!,which gives 0 as remainder.