Question

I wrote down 3 different positive whole numbers that add 96. the sum of any 2 is divisible by the third.

What is the largest of these numbers?

Answer 1

This question is really simple if you focus on this part

“The sum of any two is divisible by the third”

⟹⟹ Pairwise, the numbers must have a GCD greater than one.

Let this GCD be xx.

The first three easiest numbers I can think of are x,2xx,2x and 3x3x

Because

gcd(x,2x)=gcd(2x,3x)=gcd(3x,x)=xgcd(x,2x)=gcd(2x,3x)=gcd(3x,x)=x

The numbers sum to 9696

⟹x+2x+3x=96⟹x+2x+3x=96

⟹6x=96⟹6x=96

⟹x=16⟹x=16

The largest number is 3x=3×16=483x=3×16=48

Thanks for the A2A