How Many Three-Digit Integers are not Divisible by 3?

Question: How many three-digit integers are not divisible by 3?

1. 599
2. 600
3. 601
4. 602
5. 603

“How many three-digit integers are not divisible by 3?” –  is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. To solve GMAT Problem Solving questions a student must have knowledge about a good number of qualitative skills. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.

Approach 1

It is asked in the question that how many three-digit integers are not divisible by 3?
Firstly we need to find out the total number of three-digit numbers.
Total 3 digit numbers = 999 - 100 + 1 = 900
Therefore we got the total number of 3 digit numbers.
Now we have to find out the multiples of 3 in the range 100 - 999
The highest 3 digit multiple of 3 is 999
The smallest 3 digit multiple of 3 is 102
Total number of three digit numbers which are divisible by 3 is
(999 - 102) / 3 + 1 = 300
We are asked in the question to find out three-digit integers that are not divisible by 3.
Total number of 3 digit numbers which are not multiples of 3 = 900 - 300 = 600
600 is the correct option.

Approach 2

It is asked in the question that how many three-digit integers are not divisible by 3?
Total Single digit No. = 9 (1 to 9)
Total Two digits No. = 90 (10 to 99)
Total Single digit No. = 900 (100 to 999)
Total No. divisible by 3 from 1 through 999 = 999/3 = 333
Total No. divisible by 3 from 1 through 99 = 99/3 = 33
Total No. divisible by 3 from 100 through 999 = 333-33 = 300
So the count of numbers that are not divisible by 3 = 900-300 = 600
The correct answer will be option B.

Approach 3

First, we will find how many are actually divisible.
102 is the minimum one - it's the 34th multiple of 3.
999 is the maximum one - it's the 333 multiple of 3.
333-34 +1 (inclusive counting) = 300 numbers are divisible by 3.
Now, we have 999 total numbers. We will exclude the non 3 digit ones (from 1 to 99)
999-99=900
900-300=600
Hence, the correct answer is B.

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