3

Part of 2016 Dutch Mathematical Olympiad

Problems(1)

gcd(a, b) = gcd(a, c) = gcd(b, c) = 1 and a,b,c \ (a + b + c)

Source: Dutch NMO 2016 p3

Find all possible triples

(a, b, c)

of positive integers with the following properties: •

gcd(a, b) = gcd(a, c) = gcd(b, c) = 1

a

is a divisor of

a + b + c

b

is a divisor of

a + b + c

c

is a divisor of

a + b + c

gcd(x,y)

is the greatest common divisor of

x

y

greatest common divisorGCDnumber theorydivisordivides