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In mathematics, namely ring theory, a k-th root of unity modulo n for positive integers k, n ≥ 2, is a root of unity in the ring of integers modulo n, that is, a solution x to the equation x k ≡ 1 {\displaystyle x^{k}\equiv 1{\pmod {n}}}. If k is the smallest such exponent for x, then x is called a primitive k-th root of unity modulo n. See modular arithmetic for notation and terminology.
Do not confuse this with a primitive root modulo n, which is a generator of the group of units of the ring of integers modulo n. The primitive roots modulo n are the primitive φ {\displaystyle \varphi } -roots of unity modulo n, where φ {\displaystyle \varphi } is Euler's totient function.