1988 AIME Problems/Problem 9
Find the smallest positive integer whose cube ends in .
A little bit of checking tells us that the units digit must be 2. Now our cube must be in the form of ; using the binomial theorem gives us . Since we are looking for the tens digit, we get . This is true if the tens digit is either or . Casework:
- : Then our cube must be in the form of . Hence the lowest possible value for the hundreds digit is , and so is a valid solution.
- : Then our cube is . The lowest possible value for the hundreds digit is , and we get . Hence, since , the answer is
and . due to the last digit of . Let . By expanding, .
By looking at the last digit again, we see , so we let where . Plugging this in to gives . Obviously, , so we let where can be any non-negative integer.
Therefore, . must also be a multiple of , so . Therefore, the minimum of is .
Let . We factor an out of the right hand side, and we note that must be of the form , where is a positive integer. Then, this becomes . Taking mod , , and , we get , , and .
We can work our way up, and find that , , and finally . This gives us our smallest value, , so , as desired. - Spacesam
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