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24th Hellenic Mathematical Olympiad

Greece number theory

Problem

If the number , where is an integer, is a multiple of , find: (i) The form of the integer , (ii) The remainder of the division of with .
Solution
(i) Let , . Then , whence must divide . Since , we get that . Hence , , and therefore

(ii) We have Thus the remainder is .
Final answer
(i) ν = 11k + 2 for integer k. (ii) The remainder is 5.

Techniques

Inverses mod nPolynomials mod pGreatest common divisors (gcd)