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Modular Arithmetic

From units digits to Euler's Theorem.

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Suppose Matt has \(n\) pieces of candy that he is trying to split between 5 people equally. However, after splitting them into 5 equal piles, 2 pieces are left over. If Matt has between 0 and 100 pieces of candy, how many possible values of \(n\) are there?

What is \[\binom{12}{3}\pmod{12}?\]

Note: \(\binom{n}{k}\) denotes the binomial coefficient \[\binom{n}{k}=\frac{n!}{k!(n-k)!}.\]

Lily is buying a lottery ticket that requires a 4-digit number. She decides to choose a number by taking a random 2 digit integer \(n\), and using the number \(7^n\pmod{10000}\) as the lottery number. What is the probability that the sum of the last two digits of her ticket number sum to 7?

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