Introducing a Modular Remainder Puzzle A challenge is presented to find the smallest positive integer that satisfies specific remainder conditions: a remainder of 1 when divided by 3, 4 when divided by 5, and 6 when divided by 7. The problem is set up as an invitation to explore unexpected patterns in numbers. It immediately frames the puzzle around the interplay of division and remainders.
Translating Conditions into Congruences The approach involves transforming division remainder statements into congruence expressions. An example is reinterpreting '25 divided by 3 leaves a remainder of 1' as the equation X = 3k + 1. This translation into modular arithmetic sets the stage for systematic problem solving.
Constructing the System with Modular Equations The remainder conditions are organized into a system: X is congruent to 1 mod 3, 4 mod 5, and 6 mod 7. The system capitalizes on the fact that these moduli are pairwise relatively prime. This formation ensures that the Chinese Remainder Theorem can be safely applied to find the solution.
Solving the System through Successive Substitution Starting from X = 3k + 1, the expression is substituted into the modulo 5 condition to solve for k. After isolating k, it is rewritten as 5l + 1, which transforms X into 15l + 4. A subsequent substitution into the modulo 7 condition leads to determining l and yields the general form X = 105m + 34.
Deriving the Unique Minimal Solution By setting the free integer parameter m to zero in the final expression X = 105m + 34, the smallest positive integer solution is derived as 34. The result is unique within the cyclic pattern defined by the product of the moduli. This elegant process highlights the power of modular arithmetic and the Chinese Remainder Theorem in unlocking complex numerical puzzles.