How to solve $5x+9y=n$?

Question

Show that $n_0 = 32$ is the smallest value of $n$ for which the equation $5x+9y=n$ has a solution in $(\mathbb N \cup \{0\})^2$ for all $n \geq n_0$.

Wouldn't $(x,y)=(1,1)$ be a valid solution for $n=14$?

The example that appears before this question consider $(x,y)=(1,1)$ as a solution for $3x+5y=n$ in $(\mathbb N \cup \{0\})^2$ for $n \geq 8$.

Welcome to Mathematics Stack Exchange. Are you asked for the smallest value of $n_\color{red}0$? — J. W. Tanner, Nov 27 '20 at 19:22
Because there is no solution to $5x+9y = 16 \geq 14$, so 14 isn't the smallest number that satisfies the conditions. — Calvin Lin, Nov 27 '20 at 19:24
Thanks. Yes, the lowest value, I didn't realize that for 15, 16, ... there was no solution. — sant, Nov 27 '20 at 19:27
There is a solution for $15$ -- $(x,y)=(3,0)$ -- but not for $16$ — J. W. Tanner, Nov 27 '20 at 19:28
Yeah, true, just wanted to say that for n greater than 14 there are values with no solution. — sant, Nov 27 '20 at 19:33
That's more complicated than this question in my book, but thanks, I'll try to understand what is said there. — sant, Nov 27 '20 at 20:27

J. W. Tanner · Accepted Answer · 2020-11-27T19:36:04.920

1

$(x,y)=(1,1)$ is a valid solution for $n=14$, but I think the question asks for the smallest $n_0$ such that $5x+9y=n$ has a soution for all $n\ge n_0$, and there's no solution for $5x+9y=16>14$ in non-negative integers.

Addendum to answer OP's additional question:

$31$ cannot be expressed as $5x+9y$ with non-negative integers $x,y$, but

$32=5(1)+9(3)$, $33=5(3)+9(2)$, $34=5(5)+9(1)$, $35=5(7)+9(0)$,

and $36=5(0)+9(4)$,

and if $n-5$ can be expressed as $5x+9y$ with non-negative integers then $n$ is $5(x+1)+9y$.

edited Nov 27 '20 at 19:36

answered Nov 27 '20 at 19:22

J. W. Tanner

60,406

Ahh, I see. But how do I show that? I don't understand the example. – sant Nov 27 '20 at 19:25
Are you asking how you show that $5x+9y=16$ has no solutions in non-negative integers? – J. W. Tanner Nov 27 '20 at 19:26
No, how do I show that 32 is the smallest value that satisfies the conditions. – sant Nov 27 '20 at 19:28
Can you show there's not a solution for $31$? – J. W. Tanner Nov 27 '20 at 19:32
That tells you the smallest value is at least $32$, and my addendum shows it's not more than $32$ – J. W. Tanner Nov 27 '20 at 19:37
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Thank you so much! – sant Nov 27 '20 at 19:40

How to solve $5x+9y=n$?

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