Least common multiple

2 articles

Number Theory & Arithmetic High SchoolExplainer9 min

2519, and the Shift That Turns Nine Congruences Into One

Every one of the nine conditions leaves a remainder one short of its divisor, so x plus one is divisible by all of 2 through 10 and the answer is 2519. Minimality comes free, and the whole solution set is 2520k minus 1. The article carries the coprimality caveat, the near miss 209 that satisfies six of the nine, and a variant where no shift exists.

Number Theory & Arithmetic High SchoolExplainer9 min

Two Bells, 12 and 15 Seconds Apart, Meet at a Full Minute

Multiplying four rings a minute by five gives a quantity in inverse minutes squared, so a dimension check kills the twenty-second answer before any number theory. Converting to gaps makes the ring times two arithmetic progressions whose intersection is the least common multiple. Shift one bell by a second and the two never coincide at all.