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The function ax+gcd(a,x)+lcm(a,x)

Source: South African MO 2014 Q4

October 25, 2014
functionnumber theoryleast common multiplenumber theory unsolved

Problem Statement

(a) Let a,x,ya,x,y be positive integers. Prove: if x≠yx\ne y, the also ax+gcd⁡(a,x)+lcm(a,x)≠ay+gcd⁡(a,y)+lcm(a,y).ax+\gcd(a,x)+\text{lcm}(a,x)\ne ay+\gcd(a,y)+\text{lcm}(a,y).
(b) Show that there are no two positive integers aa and bb such that ab+gcd⁡(a,b)+lcm(a,b)=2014.ab+\gcd(a,b)+\text{lcm}(a,b)=2014.
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