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=== {{math | ''λ''(''n'')}} divides {{math | ''φ''(''n'')}} ===
This follows from elementary [[group theory]], because the exponent of any [[finite group]] must divide the order of the group. {{math | ''λ''(''n'')}} is the exponent of the multiplicative group of integers modulo {{mvar | n}} while {{math | ''φ''(''n'')}} is the order of that group. In particular, the two must equal in the cases where the multiplicative group is cyclic due to the existence of a [[primitive_root_modulo_n|primitive root]], which is the case for odd prime powers.
We can thus view Carmichael's theorem as a sharpening of [[Euler's theorem]].
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