# Prove or disprove the inequality

Let $d,\ a,\ x,\ b,\ y$ be integers. $d$ divides $a$ and $b$.

The question is: Assume $ax + by \gt 0$. Prove or disprove : $d \le ax + by$

I know that $d | ax+by$, but I can't figure out the proof for why $d$ would be $\le ax+by$.

• Is it because d = GCD(a,b) and that will always be less than a and b? Oct 5, 2014 at 4:09
• Would the statement be true if we stuck with natural numbers? Keep in mind we're working with integers. Oct 5, 2014 at 4:10
• ^In response to my above comment.... combine this with your assumption "$ax+by>0$. can you come up with counterexamples if this isn't true? Oct 5, 2014 at 4:12
• No. $a | (-a)$ and $(-a) | a$. Oct 5, 2014 at 4:15

$d| (ax+by)$ so that $$ax+by=dn$$ If $ax+by >0$, then $$n\neq 0,\ ax+by=|dn| \geq d$$