# Express any number greater than 35 as sum of x 5s and y 9s

#### dcwl

How can we prove that any number greater than or equal to 35 as a sum of x 5s and y 9s
Probably by induction

#### DarnItJimImAnEngineer

y = 0: 35, 40, 45, 50, 55, 60, â€¦
y = 1: 39, 44, 49, 54, 59, â€¦
y = 2: 38, 43, 48, 53, 58, â€¦
y = 3: 37, 42, 47, 52, 57, â€¦
y = 4: 36, 41, 46, 51, 56, â€¦

That's all integers >= 35.
You could also probably build a more formal proof using $$\displaystyle mod 5$$, but the above should be sufficient.

#### mrtwhs

31 is the largest non expressible number. Any number greater than or equal to 32 can be expressed. The proof uses something called strong induction.