2021 AIME II Problems/Problem 13
Find the least positive integer for which is a multiple of .
1000 divides this expression iff 8, 125 both divide it. It should be fairly obvious that ; so we may break up the initial condition into two sub-conditions.
(1) . Notice that the square of any odd integer is 1 modulo 8 (proof by plugging in , , , into modulo 8), so the LHS of this expression goes "5,1,5,1, ...", while the RHS goes "1,2,3,4,5,6,7,8,1, ...". The cycle length of the LHS is 2, RHS is 8, so the cycle length of the solution is . Indeed, the n that solve this equation are exactly those such that .
(2) . This is extremely computationally intensive if you try to calculate all , so instead, we take a divide-and-conquer approach. In order for this expression to be true is necessary; it shouldn't take too long for you to go through the 20 possible LHS-RHS combinations and convince yourself that or .
With this in mind we consider modulo 25. By the generalized Fermat's little theorem, , but we already have n modulo 20! Our calculation is greatly simplified. The LHS cycle length is 20, RHS cycle length is 25, the lcm is 100, in this step we need to test all the numbers between 1 to 100 that or . In the case that , , and RHS goes "3,23,43,63,83"; clearly . In the case that , by a similar argument, .
In the final step, we need to calculate modulo 125. The former is simply ; because modulo 125, . is . This time, LHS cycle is 100, RHS cycle is 125, so we need to figure out what is n modulo 500. It should be .
Put everything together. By the second subcondition, the only candidates < 100 are . Apply the first subcondition, n=797 is the desired answer.
We have that , or and by CRT. It is easy to check don't work, so we have that . Then, and , so we just have and . Let us consider both of these congruences separately.
First, we look at . By Euler's Totient Theorem (ETT), we have , so . On the RHS of the congruence, the possible values of are all nonnegative integers less than and on the RHS the only possible values are and . However, for to be we must have , a contradiction. So, the only possible values of are when .
Now we look at . Plugging in , we get . Note, for to be satisfied, we must have and . Since as , we have . Then, . Now, we get . Using the fact that , we get . The inverse of modulo is obviously , so , so . Plugging in , we get .
Now, we are finally ready to plug into the congruence modulo . Plugging in, we get . By ETT, we get , so . Then, . Plugging this in, we get , implying the smallest value of is simply . ~rocketsri
Solution 3 (Chinese Remainder Theorem and Binomial Theorem)
We wish to find the least positive integer for which Rearranging gives Applying the Chinese Remainder Theorem, we get the following systems of linear congruences: It is clear that from which we simplify to We solve each congruence separately:
- For quick inspections produce that are congruent to modulo respectively. More generally, if is odd, and if is even. As is always odd (so is ), we must have
That is, for some nonnegative integer
- For we substitute the result from and simplify:
Note that and so we apply the Binomial Theorem to the left side: Since it follows thatThat is, for some nonnegative integer
Substituting this back into we get As is a multiple of it has at least three factors of Since contributes one factor, it follows that contributes at least two factors, or must be a multiple of Therefore, such least nonnegative integer is
Finally, combining the two results from above (bolded) generates such least positive integer at ~MRENTHUSIASM (inspired by Math Jams's 2021 AIME II Discussion)
Solution 4 (Minimal Computation)
Note that and so is periodic every . Easy to check that only satisfy. Let . Note that by binomial theorem, So we have . Combining with gives that is in the form of , . Note that since Easy to check that only
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