Answer:
To prove the statement "For all integers a, b, and c, if a divides b and a divides (b^2 - c), then a divides c," we will use the definition of divisibility.
Assuming a divides b, we can write b = ka, where k is an integer.
Similarly, assuming a divides (b^2 - c), we can write b^2 - c = la, where l is an integer.
We want to show that a divides c, meaning that c = ma, where m is an integer.
First, let's consider the expression (b^2 - c). We substitute b = ka into this expression:
(b^2 - c) = (ka)^2 - c
= k^2a^2 - c
Since a divides (b^2 - c), we can rewrite this as:
k^2a^2 - c = pa
where p is some integer.
Expanding the left side of the equation, we have:
k^2a^2 - c = pa
k^2a^2 = pa + c
k^2a^2 = a(p + ka)
Now we observe that a divides the left side of the equation, k^2a^2, and since a also divides the right side of the equation, a(p + ka), we can conclude that a divides c, or c = ma, where m = p + ka, an integer.
Therefore, if a divides b and a divides (b^2 - c), then a divides c, completing the proof.