How many Integers Less than 1000 have no Factors (other than 1) GMAT Problem Solving

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

1. 400
2. 401
3. 410
4. 420
5. 421

Correct Answer: A

Approach Solution (1):

First of all, it should be “how many positive integers less than 1000 have no factors (other than 1) in common with 1000”, as if we consider negative integers answers will be: infinitely many.

1000 = \(2^3*5^3\)so basically we are asked to calculate the # of positive integers less than 1000, which are not multiples of 2 or/ and 5.

Multiples of 2 in the range 0 – 1000, not inclusive – \(\frac{998-2}{2}+1=499\) ;

Multiples of 5 in the range 0 – 1000, not inclusive – \(\frac{995-5}{5}+1=199\) ;

Multiples of both 2 and 5, so multiples of 10 – \(\frac{990-10}{10}+1=99\)

Total # of positive integers less than 1000 is 999, so # integers which are not factors of 2 or 5 equals to 999 – (499 + 199 – 99) = 400

Approach Solution (2):

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

Number of even numbers less than 1000:

\(\frac{998-2}{2}+1=499\)

Number of multiples of five less than 1000:

\(\frac{995-5}{5}+1=199\)

We must find the double-counted numbers, also 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:

\(\frac{990-10}{10}+1=99\)

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

499 + 199 – 99 = 5999

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

999 – 599 = 400

“How many integers less than 1000 have no factors (other than 1) in common with 1000?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills.

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