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# Modular Arithmetic Problem

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Find the remainder when 24^50 - 15^50 is divided by 13.

I know that we'd have to simplify 24^50 and 15^50, but it's hard to do. Can anyone help me?

May 6, 2020

#1
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Sorry, I don't know how to reduce it, but:

[24^50  -  15^50] mod 13 = 0

May 6, 2020
#2
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Find the remainder when $$24^{50} - 15^{50}$$ is divided by $$13$$.

$$\begin{array}{|rcll|} \hline && \mathbf{24^{50} - 15^{50} \pmod{13}} \quad | \quad 24 \equiv -2 \pmod{13},\ 15 \equiv 2 \pmod{13} \\ &\equiv& (-2)^{50} - 2^{50} \pmod{13} \\ &\equiv& 2^{50} - 2^{50} \pmod{13} \\ &\equiv& \mathbf{0 \pmod{13}} \\ \hline \end{array}$$

May 7, 2020