211 211 is a prime

What is the remainder when 211 ! 211! is divided by 213 213 ?


The answer is 0.

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4 solutions

Ayush Pattnayak
Oct 23, 2014

213=3*71,and these two factors are covered in 211!,which gives 0 as remainder.

211/71=2.972

Anita Paul - 6 years, 7 months ago

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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 = 24 4! = 4 \times 3 \times 2 \times 1 = 24

Sharky Kesa - 6 years, 7 months ago
Sharky Kesa
Oct 23, 2014

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

Anita Paul - 6 years, 7 months ago

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211 ! 211! is what I have written!!!

Sharky Kesa - 6 years, 7 months ago

By Wilson's Theorem ( ( n 1 ) ! 1 m o d ( n ) (n-1)! \equiv -1 \ \mod(n) ) we have:

212 ! 212 m o d ( 213 ) 212! \equiv 212 \ \mod (213)

Then 212 ! / 212 212 / 212 m o d ( 213 ) 212!/212 \equiv 212/212 \ \mod(213)

211 ! 0 m o d ( 213 ) 211! \equiv 0 \ \mod(213)

Rupkatha Ghosh
Nov 16, 2014

213=3*71,and these two factors are there in 211!,

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