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Calculating pow(a,nCr) % p
By asif599, history, 7 years ago, In English

Hello Codeforces How to calculate pow(a,nCr) % p efficiently? Here 1 <= n <= 10^6, 1 <= r <= 10^6 and 1<= a <= 10^6

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7 years ago, # |
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Sorry, i got it wrong

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7 years ago, # |
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Hint: Use Legendre's formula and Fermat's little Theorem.

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7 years ago, # |
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To find a^b %m when b is too large, we calculate a^(b%(m-1))%m this can be proceed using fermat little theorem.

To calculate nCr%(m-1) you can use chinese remainder theorem if m-1 is non prime and square free.

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    7 years ago, # ^ |
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    I prove a^b %m= a^(b%(m-1))%m in this way, but I am not sure whether it is correct or not.

    Fermat's little theorem tells that a^(m-1) %m=1, and thus we can write b=x*(m-1)+y. Then, a^b %m=a^(x*(m-1)+y) %m= a^y * (a^(m-1))^x %m= (a^y %m) * (a^(m-1) %m)^x=a^y %m =a ^ (b%(m-1)) %m.

    If this is correct, I really have never considered using Fermat's little theorem in this manner... My first reaction is to calculate the inverse of b when we need to compute a/b %m. Thank you so much for sharing such wonderful idea and extending my thought.

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    7 years ago, # ^ |
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    Can you elaborate more on how you calculte nCr%(m — 1) using chinese remainder theoreom ? Suppose that (m — 1) is non-prime.

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