Congruence Problem

Number Theory Level 2

What is the remainder of 10000! divided by 10001?


The answer is 0.

Culver Kwan by Culver Kwan , 16, Hong Kong

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1 solution

Note that 73 ∗ 137 = 10001 73*137=10001

10000 ! = 10000 ∗ 9999 ∗ . . . ∗ 137 ∗ . . . ∗ 73 ∗ . . . 1 10000!=10000*9999*...*137*...*73*...1

so when 10000 ! 10000! is divided by 10001 10001 we get :

10000 ∗ 9999 ∗ . . . ∗ 139 ∗ 138 ∗ 136 ∗ 135 ∗ . . . ∗ 75 ∗ 74 ∗ 72 ∗ 71 ∗ . . . ∗ 1 10000*9999*...*139*138*136*135*...*75*74*72*71*...*1

and the reminder is 0 0 .

5 Helpful 0 Interesting 0 Brilliant 0 Confused

Notice that you cannot use Wilson's theorem with 10001, since it is not prime (unlike 101)!

Toby M - 2 years, 5 months ago

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