Question

For all integers a and b, if a∣b then a² ∣b² .

a) True

b) False

Answer 1

If a divides b, then
`a²` divides
`b²` for all integers a and b.

Step-by-step explanation:

For all integers a and b, if a divides b (written as a∣b), then
`a²` divides
`b²` (written as a²∣b²). To prove this, let's assume that a∣b for some integers a and b. This means that there exists an integer k such that
`b = k * a`. Now we can square both sides of this equation to get
`b² = (k * a)² = k² * a²`. Since
`k² * a²` is also an integer, we can conclude that
`a²` divides
`b² (i.e., a²∣b²)`.