Loading [MathJax]/extensions/TeX/mathchoice.js

Thursday, September 21, 2023

2023/035) Prove that 4 . 29! + 5 ! \equiv 0 \pmod {31}

We have as 31 prime  using Wilsons theorem

30! + 1 \equiv 0 \pmod {31}

or 30 * 29! + 1 \equiv 0 \pmod {31}

as 30  \equiv - 1 \pmod {31}

so we have  -1 * 29! + 1 \equiv 0 \pmod {31}

Or  29! -1 \equiv 0 \pmod {31}

Or 4 * 29! - 4 \equiv 0 \pmod {31}

adding 124 which is multiple of 31 we get

4 * 29! + 120 \equiv 0 \pmod {31}

or 4 * 29! + 5! \equiv 0 \pmod {31}

Proved 

No comments: