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Please help me out in performing modulo operation on a fraction. I saw a reference to Gauss algorithm but was not able to find anything there.

example : (1/4) mod 23. What is the answer to this? And detailed step by step explanation is highly appreciated. Thanks

sbp
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  • https://math.stackexchange.com/questions/864568/is-it-possible-to-do-modulo-of-a-fraction may be a possible match – Kyan Cheung Dec 07 '17 at 10:12
  • Another possible match: https://math.stackexchange.com/questions/586595/finding-modular-of-a-fraction – cyanide Dec 07 '17 at 10:16
  • One should avoid the fractional notation in modular arithmetic, as it is rather confusional with ‘true’ fractions. For instance, there is a canonical morphism from $\mathbf Z$ to $\mathbf Z/23\mathbf Z$, but it cannot be extended to $\mathbf Q$. – Bernard Dec 07 '17 at 10:20
  • You can use MathJax to format your posts. – gen-ℤ ready to perish Dec 07 '17 at 10:36

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You need to find the multiplicative inverse of 4 modulo 23, that is, an integer $q$ with $4\cdot q \equiv 1 \pmod{23}.$ For this, use the extended Gauss algorithm to find integers $p$ and $q$ such that $23\cdot p+4\cdot q =1,$ which is possible since $\gcd(23,4)=1.$ It follows that $q=1/a \pmod{23}.$

Reiner Martin
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