{
  "aliases": [
    "fermat-euler-theorem",
    "euler-totient-theorem"
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  "body": "\n## Statement\n\nLet $n$ be a positive integer and let $a$ be an integer [[coprime|0GFKP8F8AG]] to $n$.\nThen $a^{\\varphi(n)} \\equiv 1 \\pmod n$, where $\\varphi$ is [[Euler's totient function|0VV04M41AN]].\n\n## Remarks\n\nWhen $n = p$ is prime, $\\varphi(p) = p - 1$, and the result is [[Fermat's little theorem|0M25PRN3WQ]].\nA proof uses [[Lagrange's theorem|3C53TD]]: the group $(\\mathbb{Z}/n\\mathbb{Z})^\\times$ has order $\\varphi(n)$, by the [[number of invertible residues modulo n|06RJWZCG95]].\nThe result lets us reduce exponents. If $\\gcd(a, n) = 1$ and $k \\equiv l \\pmod{\\varphi(n)}$, then $a^k \\equiv a^l \\pmod n$.\nFor example, $\\varphi(10) = 4$, so $7^{4} \\equiv 1 \\pmod{10}$, and the last digit of $7^{222}$ is the last digit of $7^2 = 49$, that is, $9$.\n\n## History\n\nEuler proved the theorem in 1763.\n\n## Explanations\n\n### L1 human\n\nPowers of $a$ modulo $n$ repeat in a cycle, and the cycle comes back to $1$.\nEuler's theorem gives one exponent that always brings $a$ back to $1$, namely $\\varphi(n)$.\nFor $n = 12$ and $a = 5$: $\\varphi(12) = 4$ and $5^4 = 625 = 52 \\cdot 12 + 1$.\n",
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