In elementary number theory, the lifting-the-exponent (LTE) lemma provides several formulas for computing the p-adic valuation of special forms of integers. The lemma is named as such because it describes the steps necessary to "lift" the exponent of in such expressions. It is related to Hensel's lemma. Background The exact origins of the LTE lemma are unclear; the result, with its present name and form, has only come into focus within the last 10 to 20 years. However, several key ideas used in its proof were known to Gauss and referenced in his Disquisitiones Arithmeticae. Despite chiefly featuring in mathematical olympiads, it is sometimes applied to research topics, such as elliptic curves. Statements For any integers , a positive integer , and a prime number such that and , the following statements hold: When is odd: If , then . If is odd and , then . When : If and is even, then . If and is odd, then . (Follows from the general case below.) Corollary: If , then and thus . For all : If and , then . If , and odd, then . Outline of proof Base case The base case when is proven first. Because , The fact that completes the proof. The condition for odd is similar. General case (odd p) Via the binomial expansion, the substitution can be used in (1) to show that because (1) is a multiple of but not . Likewise, . Then, if is written as where , the base case gives . By induction on , A similar argument can be applied for . General case (p = 2) The proof for the odd case cannot be directly applied when because the binomial coefficient is only an integral multiple of when is odd. However, it can be shown that when by writing where and are integers with odd and noting that because since , each factor in the difference of squares step in the form is congruent to 2 modulo 4. The stronger statement when is proven analogously. In competitions Example problem The LTE lemma can be used to solve 2020 AIME I #12: Let be the least positive integer for which is divisible by Find the number of positive integer divisors of . Solution. Note that . Using the LTE lemma, since and but , . Thus, . Similarly, but , so and . Since ,