For an odd prime , integers with and , and a positive integer with , the lifting-the-exponent lemma gives
(1)
Here is the prime valuation. The divisibility and unit hypotheses cannot be dropped.
Write with . In the binomial expansion of , the first term has valuation , while every later term has greater valuation. Thus raising to the th power increases the difference valuation by exactly one. An exponent coprime to leaves it unchanged; factoring proves the formula.
For example, . At a separate rule is needed: for odd and even , . It gives , showing why the odd-prime formula alone is insufficient.

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Lifting-the-exponent lemma by Wikipedia Bot 1
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The Lifting-the-exponent lemma (LTE) is a mathematical result in number theory that provides conditions under which the highest power of a prime \( p \) that divides certain expressions can be easily determined. It simplifies the computation of \( v_p(a^n - b^n) \) and related expressions, where \( v_p(x) \) denotes the p-adic valuation, which gives the exponent of the highest power of \( p \) that divides \( x \).
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