# Difference between revisions of "2020 AIME I Problems/Problem 12"

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## Problem

Let $n$ be the least positive integer for which $149^n-2^n$ is divisible by $3^3\cdot5^5\cdot7^7.$ Find the number of positive integer divisors of $n.$

## Solution

Lifting the Exponent shows that $$v_3(149^n-2^n) = v_3(n) + v_3(147) = v_3(n)+1$$ so thus, $3^2$ divides $n$. It also shows that $$v_7(149^n-2^n) = v_7(n) + v_7(147) = v_7(n)+2$$ so thus, $7^5$ divides $n$.

Now, multiplying $n$ by $4$, we see $$v_5(149^{4n}-2^{4n}) = v_5(149^{4n}-16^{n})$$ and since $145^{4} \equiv 1 \pmod{25}$ and $16^1 \equiv 16 \pmod{25}$ then $v_5(149^{4n}-2^{4n})=1$ meaning that we have that by LTE, $4 \cdot 5^4$ divides $n$.

Since $3^2$, $7^5$ and $4\cdot 5^4$ all divide $n$, the smallest value of $n$ working is their LCM, also $3^2 \cdot 7^5 \cdot 4 \cdot 5^4 = 2^2 \cdot 3^2 \cdot 5^4 \cdot 7^5$. Thus the number of divisors is $(2+1)(2+1)(4+1)(5+1) = \boxed{270}$.

## See Also

 2020 AIME I (Problems • Answer Key • Resources) Preceded byProblem 11 Followed byProblem 13 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 All AIME Problems and Solutions

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