MathDB

4

Part of 2014 South africa National Olympiad

Problems(1)

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

Source: South African MO 2014 Q4

10/25/2014
(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.
functionnumber theoryleast common multiplenumber theory unsolved