Confused! PrepSwift: Adding Numbers to Multiples

GRE Quant Problem Solving
1

a is a positive integer.

Quantity A
The number of factors of a

Quantity B
The number of factors of 2a

Answer Choices
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the information given.

I originally thought it would be answer choice D but the correct choice is B and I wondered why. If I use the positive integer like 17 for example. The only factors would be 1 and 17 resulting in 2 factors. Plugging that in for quantity b (2a)=34 giving us 1, 34, 2, and 17. Giving this example the relationship cannot be determined since it can go both ways depending on what a is defined as. Am I overthinking this problem?

2

No, you are not overthinking this. You are on the right track. Quantity B is always greater than Quantity A.

The proof that I can think of is

Every factor of a is also a factor of 2a, since a divides 2a
But 2a itself is not a factor of a (unless a = 0, which isn’t allowed here and a has to be positive).
So 2a is a new factor of 2a that isn’t in the factors list of a.

All factors of a ⊆ factors of 2a and 2a has at least one new factor (2a)

This is enough to pick B, we don’t actually need to calculate the number of factors of a and 2a.