2020 AIME I Problems/Problem 12
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Problem
Let be the least positive integer for which is divisible by Find the number of positive integer divisors of
Solution
Lifting the Exponent shows that so thus, divides . It also shows that so thus, divides .
Now, multiplying by , we see v_5(149^{4n}-2^{4n}) = v_5(149^{4n}-16^{n})145^{4} \equiv 1 \pmod{25}16^1 \equiv 16 \pmod{25}v_5(149^{4n}-2^{4n})=14 \cdot 5^4n$.
Since$ (Error compiling LaTeX. ! Missing $ inserted.)3^27^54\cdot 5^4nn3^2 \cdot 7^5 \cdot 4 \cdot 5^4 = 2^2 \cdot 3^2 \cdot 5^4 \cdot 7^5(2+1)(2+1)(4+1)(5+1) = \boxed{270}$.
See Also
2020 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
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