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PrintRomanian Mathematical Olympiad
Romania number theory
Problem
Let be an integer, . Find the remainder of the division of the number by .
Solution
We notice that . If , the residue is .
If , the residue is . If , the residue is . If , the residue is .
If , the residue is . If , the residue is . If , the residue is .
Final answer
n(n+1)(n+2) ≡ 6 (mod n−1). Hence the remainder is 6 for n ≥ 8; for n = 2,3,4,7 it is 0; for n = 5 it is 2; for n = 6 it is 1.
Techniques
Modular ArithmeticPolynomial operations