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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "6cfdb9dc-df35-4ac5-9fa9-da7cae02d2b8",
+   "metadata": {},
+   "source": [
+    "### Euler's product formula\n",
+    "\n",
+    "$\\phi(n)=n\\prod_{p|n}\\left(1-\\frac{1}{p}\\right)$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0be1d436-5075-4f2f-96ec-e225179209a7",
+   "metadata": {},
+   "source": [
+    "### Euler's theorem\n",
+    "\n",
+    "$gcd(a, n)=1$\n",
+    "\n",
+    "$a^{\\phi(n)}\\equiv1 \\bmod n$"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "SageMath 10.6",
+   "language": "sage",
+   "name": "sagemath-10.6"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.12.5"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}