swerve wrote:How many integers less than 1000 have no factors (other than 1) in common with 1000?

A. 400
B. 399
C. 410
D. 420

The OA is A

Source: GMAT Paper Tests

The question means positive integers less than 1000; if we consider negative integers as well, the answer would be an infinite number of integers.

Let's factorize 1000.

1000 = 2^3*5^3. So, 1000 have two prime factors: 2 and 5.

So, we have to find out less than 1000 positive integers that do not have a factor 2 or 5.

Let's do another way. Let's find out the integers that have at least one common factor with 1000 and then exclude them to get the answer.

Multiples of 2 that are within 1-999, inclusive = 1000/2 - 1 = 499 integers;
Multiples of 5 that are within 1-999, inclusive = 1000/5 - 1 = 199 integers;

Total number of such integers = 499 + 199 = 698 integers

Note that in the count of 499, there would be a few integers that are multiples of 5 as well. We must exclude them as they are counted in 199 we well. Those multiples are multiples of 2*5 = 10.

Multiples of 10 that are within 1-999, inclusive = 1000/10 - 1 = 99 integers;

Number of integers that have at least one common factors with 1000 = 698 - 99 = 599

Thus, the integers less than 1000 have no factors (other than 1) in common with 1000 = 999 - 599 = 400

The correct answer: A

Hope this helps!

-Jay
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Total number of required integers = 598 - 99 = 399

swerve wrote:How many integers less than 1000 have no factors (other than 1) in common with 1000?

A. 400
B. 399
C. 410
D. 420

The OA is A

Source: GMAT Paper Tests

Since 1,000 breaks down to prime factors of twos and fives, we need to find all the numbers less than 1,000 that do not contain those factors. To do so, let's find all the numbers less than 1000 that contain factors of twos and fives. Note that all even numbers (multiples of 2) and all multiples of 5 must be accounted for.

Number of even numbers less than 1000:

(998 - 2)/2 + 1 = 499

Number of multiples of five less than 1000:

(995 - 5)/5 + 1 = 199

We must find the double-counted numbers, also called overlap numbers, which are numbers that are multiples of both 2 and 5. To find the overlap, we need to determine the number of multiples of 5 and 2 (or of 10) less than 1000:

(990 - 10)/10 + 1 = 99

Thus, the number of multiples of 2 or multiples of 5 less than 1000 is:

499 + 199 - 99 = 599

Finally, the number of numbers less than 1000 that ARE NOT multiples of 2 or 5 is:

999 - 599 = 400

Answer: A