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lcm 2 player game a = lcm (x, y), b = lcm(x, z), c = lcm(y, z)

Source: Tournament of Towns, Senior A-Level Paper, Spring 2020 , p2

June 3, 2020

LCMnumber theoryleast common multiple

Problem Statement

Alice had picked positive integers

a, b, c

and then tried to find positive integers

x, y, z

such that

a = lcm (x, y)

b = lcm(x, z)

c = lcm(y, z)

. It so happened that such

x, y, z

existed and were unique. Alice told this fact to Bob and also told him the numbers

a

and

b

. Prove that Bob can find

c

. (Note: lcm = least common multiple.)
Boris Frenkin