Maths Olympiad Prep

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Number theory Difficulty 5.6 AIME, harder Find the answer

7. Find a complete residue system r1,,r4r_{1}, \cdots, r_{4} modulo 4 and a complete residue system s1,,s5s_{1}, \cdots, s_{5} modulo 5, such that \square
(i) risj(1i4,1j5)r_{i} s_{j}(1 \leqslant i \leqslant 4,1 \leqslant j \leqslant 5) is a complete residue system modulo 20;
(ii) ri+sj(1i4,1j5)r_{i}+s_{j}(1 \leqslant i \leqslant 4,1 \leqslant j \leqslant 5) and risj(1i4,1j5)r_{i} s_{j}(1 \leqslant i \leqslant 4,1 \leqslant j \leqslant 5) are simultaneously complete residue systems modulo 20.

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

7. Take ri=5i+4,1i4,sj=5+4j,1j5r_{i}=5 i+4,1 \leqslant i \leqslant 4, s_{j}=5+4 j, 1 \leqslant j \leqslant 5 so that (i) and (ii) both hold.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.