Final answer:
To prove that a divides b and c implies a divides the difference b-c, we use the definition of division to express b and c as multiples of a, which upon subtraction shows that a indeed divides their difference.
Step-by-step explanation:
Proving a|b and a|c implies a | (b-c) To prove that for all integers a, b, and c, if a|b and a|c then a | (b-c), we start by understanding the notation. The notation a|b means 'a divides b', which implies there exists an integer k such that b = ak. Similarly, a|c means there exists an integer m such that c = am.
Now, considering the expression b - c, we substitute b and c with their equivalent expressions involving a:
Therefore, b - c = ak - am.
Factoring out a from both terms, we get (b - c) = a(k - m).
Since k and m are integers, their difference (k - m) is also an integer. Let's denote it as n, hence (b - c) = an.
This shows that a divides (b - c), as n is an integer, so we have proved the initial statement:
If a|b and a|c, then a | (b-c).