Final answer:
To prove that if integers a divides b and b divides a, it implies a = ±b, we used fundamental properties of integer division and multiplication. We found that the only integers that multiply together to give 1 are ±1, leading to the conclusion that a = ±b.
Step-by-step explanation:
We have been asked to prove that if a and b are nonzero integers such that a divides b (a | b) and b divides a (b | a), then this is equivalent to saying that a = ±b.
Firstly, a | b means there exists an integer k such that b = ka. Similarly, b | a means there exists an integer m such that a = mb. If we substitute the first into the second, we get a = m(ka), which simplifies to a = mk·a. Since a is not zero, we can divide both sides by a to get 1 = mk.
For the integers, the only way for the product of two numbers m and k to be 1 is if both are 1 or both are -1. This means m and k are each either 1 or -1. Therefore, b = ±a, which leads us to a = ±b, proving the original statement.
The properties of integer division and multiplication we used here are fundamental concepts in mathematics, ensuring that the result is always an integer and the sign rules are applied correctly.