# Calculate fractional part in $\displaystyle \left\{\frac{22^{56}}{17}\right\}$

How can I calculate the fractional part $$\displaystyle \left\{\frac{22^{56}}{17}\right\}$$

Here $\displaystyle \{x\} = \text{ fractional part of$x$}$

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## 1 Answer

Hint: Take $22^{56}$ modulo $17$ by using Fermat's Little Theorem.

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Thanks Joe Zeng Got it. Fermat's Little Theorem.$a^{p-1}=a mod(p)$ where $a\;,p\in \mathbb{Z^{+}}$ and $a,p$ are relatively prime $(22)^{16} = 1mod(p)\Leftrightarrow (22)^{48}=1mod(p)$ and $(22)=5mod(17)\Leftrightarrow (22)^8 =(5)^8mod(17)=(8)^4mod(17)=(13)^2mod(17)=16mod(17)$ so $(22)^{56}=16mod(17)$ – juantheron Feb 17 '13 at 15:34
That is correct. – Joe Z. Feb 17 '13 at 21:00