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Fermat’s Little Theorem.md

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+#Math #NT
+
+# Fermet’s Little Theorem
+
+If $p$ is a prime integer:
+
+$$
+a^{p - 1} \equiv 1 \mod p \\
+a^p \equiv a \mod p
+$$
+$$
+a^{p - 1} \equiv 1 \mod p \\
+a^p \equiv a \mod p
+$$
+
+# Proof
+
+Let $p$ be a prime integer. Say a necklace has $p$ beads and $a$ possible colors per bread. Except for a necklace with only one color, each combination of necklace colors has $p$ permutations. Therefore:
+
+$$
+a^p \equiv a \mod p
+$$