i did use my poof to prove. im not sure im doing right.
here is exercises:
'Suppose a,b,c are integers if a|b and a|c, then a|(b+c).'
my answer:
Proof. Suppose a|b ans a|c.
By Definition of divisibility. we know a|b means there is an integer x with b= ax.
Likewise, a|c means there is an integer y for which c= ay
Thus b+c=(ax+ay)= a(x+y)
therefore a|(b+c).
Looks good to me.
thank you..
Your solution looks correct! You have correctly used the definitions of divisibility to prove that if a divides both b and c, then it must also divide their sum (b+c). Your proof shows the necessary steps and reasoning, so well done!
If you want to double-check your work, here's a breakdown of how you arrived at your conclusion:
1. Start by assuming a|b and a|c. This means there exist integers x and y such that b = ax and c = ay.
2. Next, consider the sum of b and c. It can be written as b + c = (ax) + (ay) = ax + ay.
3. Factor out the common factor of a from the expression ax + ay. We get b + c = a(x + y).
4. Since (x + y) is an integer, let's call it z for simplicity. So, b + c = az.
5. By substitution, we see that a divides (b + c) since there exists an integer z (x + y = z) such that (b + c) = az.
Therefore, you have successfully proven that if a divides both b and c, then it must also divide their sum (b + c). Keep practicing and you'll become more confident in your proof-writing skills!