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