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Does positive integer z equal 5?

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by brentvwilliams » Mon Oct 23, 2017 6:27 am
Can take a shot...

Does positive integer z equal 5?

(1) z is not a factor of 42.

Factors of 42: 1,2,3,6,7,14,21, 42.
z is not equal to any of these numbers.
This does not tell us anything else about z. It could be any other positive integer. Insufficient.

(2) 5 is the largest divisor of z.

Since 5 is a prime number, and the only factors of a prime number are one and itself.
This means that if the largest divisor (factor) of z is 5, then z must be equal to 5. Sufficient.
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by ErikaPrepScholar » Mon Oct 23, 2017 10:27 am
To expand on the comment above, let's look at some other options for Statement 2.

We know that 5 is the greatest factor of z. So what are some numbers that have 5 as a factor? We already established that 5 is the greatest factor of 5. But what about 10 or 15 or 20? Could another number have 5 as its greatest factor?

The factors of 10 are 1, 2, 5, and 10.
The factors of 15 are 1 , 3, 5, and . 15.
The factors of 20 are 1, 2, 4, 5, 10, and 20.

We need to recognize that a number's greatest factor will always be itself, whether or not it is prime. So all numbers greater than 5 that have 5 as a factor will also have another greater factor. This means that 5 is the *only number* with 5 as its greatest factor, making statement 2 sufficient.
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