A number theory problem by Gautam Arya

Number Theory Level 4

The equation $3x + 5y = 53$ has exactly 4 pairs of positive integer solutions: $(x,y) = (1,10), (6,7), (11,4), (16,1)$ .

Find the largest possible value of $k$ such that the equation $3x + 5y = k$ has exactly 4 pairs of positive integer solutions $(x,y)$ .

1 solution

Giorgos K.
Feb 24, 2018

I just searched the first 1000 k on Mathematica

Select[Range@1000,Length@Solve[3x+5y==#&&x>0&&y>0,{x,y},Integers]==4&]

which returns all the k with exactly 4 pairs of positive integer solutions

{53, 56, 58, 59, 61, 62, 63, 64, 65, 66, 67, 69, 70, 72, 75}