Science:Math Exam Resources/Courses/MATH312/December 2010/Question 03
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Find all three-digit combinations that can occur as the last three digits of where a ranges over the integers. Give a simple criterion for predicting which of these possibilities occurs for a given a without computing . Prove that your criterion is correct.
Hint: Investigate modulo 8 and modulo 125 separately, then use the Chinese Remainder Theorem.
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We proceed as in the hints. Modulo 8, we have that if a is even, then the residue of this number is 0 modulo 8. If it is odd, then Euler's Theorem tells us that since , we have
Similarly, modulo 125 we have that if 5 divides a, then the residue is 0 and otherwise, Euler's Theorem tells us that since , we have
This gives us four possible systems. Either
As 8 and 125 are coprime, we have via the Chinese Remainder Theorem that there exists a unique solution to each of these system of equations modulo 1000 (the product of 8 and 125). We break this down into cases.
Case 1: . This case occurs when 10 divides a. In this case, by inspection, we can see that the last three digits here are given by 000 since 0 is the unique solution to this system.
Case 2: . This occurs when a is odd and divisible by 5. In this case, we have that and so . The inverse of 8 modulo 125 is given by the Euclidean algorithm:
and back substituting gives
so the inverse is 47. Thus
so . Thus recalling that we have that
and so the last 3 digits are 625 in this case.
and so the last 3 digits are 376 in this case.
Case 4: . This occurs when a is odd and not divisible by 5. In this case, we have that and so . The inverse of 8 modulo 125 as computed above is 47. Thus and hence . Thus recalling that we have that
and so the last 3 digits are 001 in this case. This completes all cases.