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2020 AIME I Problems/Problem 12

Revision as of 21:52, 13 March 2020 by Fireflame241 (talk | contribs) (→‎Solution 1)

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 1

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 $n$.

The divisibility criterion implies $149^n\equiv 2^n \pmod{5}$. This only occurs when $4 | n$, so let $n=4m$. We see \[5 = v_5(149^{4m}-2^{4m}) = v_5((149^4)^{m}-(2^4)^{m}) = v_5(149^4-2^4)+v_5(m)\] Since $149^{4} \equiv 1 \pmod{25}$ and $16^1 \equiv 16 \pmod{25}$, then $v_5(149^4-2^4)=1$, so $v_5(m)=5$, hence $4 \cdot 5^4$ 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 $(2+1)(2+1)(4+1)(5+1) = \boxed{270}$.

~kevinmathz + fireflame241

Solution 2 (Simpler, just basic mods and Fermat's theorem)

Note that for all n, $149^n - 2^n$ is divisible by 1$49-2 = 147$ because that is a factor. That is $3\cdot7^2$, so now we can clearly see that the smallest $n$ to make the expression divisible by $3^3$ is just $3^2$. Similarly, we can reason that the smallest n to make the expression divisible by $7^7$ is just $7^5$.

Finally, for $5^5$, take mod $5$ and mod $25$ of each quantity (They happen to both be $-1$ and $2$ respectively, so you only need to compute once). One knows from Fermat's theorem that the maximum possible minimum $n$ for divisibility by $5$ is $4$, and other values are factors of $4$. Testing all of them (just $1,2,4$ using mods-not too bad), $4$ is indeed the smallest value to make the expression divisible by $5$, and this clearly is NOT divisible by $25$. Therefore, the smallest $n$ to make this expression divisible by $5^5$ is $2^2 \cdot 5^4$.

Calculating the LCM of all these, one gets $2^2 \cdot 3^2 \cdot 5^4 \cdot 7^5$. Using the factor counting formula, the answer is $3\cdot 3\cdot 5\cdot 6$ = $\boxed{270}$.

-Solution by thanosaops

See Also

2020 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 11 		Followed by
Problem 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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