Number Properties. Third Edition. Page 181, Question 20

[jeffhicks05]

This is just a terminology problem. Here is the question.

x=120, y=150.

Quant A: The number of positive divisors of x.

Quant B: The number of positive divisors of y.

The answer lists the factors of each number, and claims that quantity A is larger because 120 has more factors than 150, implying that there are no divisors that are not factors.

However, a divisor can be an integer that is not a factor (to my understanding). That is, when a factor acts as the divisor, the remainder is 0. When a divisor is not a factor, the remainder is non-zero. I took this implied definition from the books section on remainders.

By this definition of a divisor, the two quantities are equal, because the divisor can be any positive integer [1, infinity).

Where am I going wrong in my reasoning? Thanks!

Jeff

[tommywallach]

Hey Jeff,

The primary definition of divisor is "without a remainder." The other usage of "divisor" would not apply here.

The definition you're thinking of only applies to a pre-existing operation, in which case the divisor is defined. For example, if you saw:

2/3

You could say that 3 is the "divisor" here, because it is the thing that will be dividing into something else.

But if a question ever asked how many "divisors" a certain number had, the word would always be synonymous with "factors". The other definition wouldn't apply, because there is no known, defined thing that is dividing into something else.

Hope that helps!

-t