A student is proving the following by contradiction: “If a n...

Question

A student is proving the following by contradiction:

“If a number is irrational, then it cannot be expressed as a fraction of two integers.”

They begin:

1. Assume the number is irrational and it can be written as a fraction , where , and is in lowest terms.

2. Multiply both sides by to get:

3. [Missing Step]

4. This leads to a contradiction, because must then also be irrational, contradicting that is an integer.

Which is the correct missing step to complete the proof?

A

Since

is rational, multiplying it by an irrational number gives a rational number.

B

Since

is an integer, the product

is irrational.

C

Since

is in lowest terms,

and

share no common factor.

D

Because irrational numbers can be written as square roots,

must be irrational.