Smallest Integer with Remainder Rules

This interview question presents a number theory puzzle about an integer constrained by a sequence of remainder conditions under several different divisors. The setup is a classic pure mathematics problem about congruences: an unknown positive integer behaves in a structured way when divided by each of a run of small integers. Candidates are asked to recognize and exploit the pattern in those remainders, relate the conditions to a simpler equivalent requirement, and then identify the smallest integer that fits all of them at once.

The solution leans on modular arithmetic, least common multiples, and recognizing when a system of congruences can be collapsed to a single divisibility condition. Strong answers use clear notation to express the congruence pattern, then systematically reduce the problem instead of trial-and-error search. Interviewers look for comfort with divisibility, the ability to generalize from small cases to a rule, and logical justification for why the constructed number is indeed the smallest possible satisfying all the given remainder rules.

What it tests

This problem class is governed by the structure of simultaneous congruences where each modulus's remainder is exactly one less than the modulus itself. The key insight is that if a number leaves a remainder of $d-1$ when divided by $d$, then adding 1 to that number makes it divisible by $d$. Thus, the problem reduces to finding a number that is one less than a common multiple of all the divisors in question. The smallest such number is the least common multiple (LCM) of the divisors, minus one. This principle holds because the LCM is the smallest number that is divisible by each divisor, ensuring all congruence conditions are satisfied simultaneously.