Proof pearl: Bounding least common multiples with triangles

Abstract

We present a proof of the fact that 2n ≤ lcm{1, 2, 3, …, (n+1)}. This result has a standard proof via an integral, but our proof is purely number theoretic, requiring little more than list inductions. The proof is based on manipulations of a variant of Leibniz’s Harmonic Triangle, itself a relative of Pascal’s better-known Triangle.