2018 AIME I Problems/Problem 11

Find the least positive integer $n$ such that when $3^n$ is written in base $143$ , its two right-most digits in base $143$ are $01$ .

Solutions

Modular Arithmetic Solution- Strange (MASS)

Note that $3^n \cong 1 (\mod 143^2)$ . And $143=11*13$ . Because $gcd(11^2, 13^2) = 1$ , $3^n \cong 1 (\mod 121 = 11^2)$ and $3^n \cong 1 (mod 169=13^2)$ .

If $3^n \cong 1 (\mod 121)$ , one can see the sequence $1, 3, 9, 27, 81, 1, 3, 9...$ so $n \mid 5$ .

Now if $3^n \cong 1 (\mod 169)$ , it is harder. But we do observe that $3^3 \cong 1 (\mod 13)$ , therefore $3^3 = 13a + 1$ for some integer $a$ . So our goal is to find the first number $p_1$ such that $(13a+1)^(p_1) \cong 1 (\mod 169)$ . In other words, the $a$ coefficient must be $0 (\mod 169)$ . It is not difficult to see that this first $p_1=13$ , so ultimately $3^39 \cong 1 (\mod 169)$ . Therefore, $n \mid 39$ .

The first $n$ satisfying both criteria is $5*39=\boxed{195}$ .

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2018 AIME I Problems/Problem 11

Solutions

Modular Arithmetic Solution- Strange (MASS)