How Many Integers from 1 to 100 are not Divisible by 2, 3, and 5? GMAT Problem Solving

Question: How many integers from 1 to 100 are not divisible by 2, 3, and 5?

1. 26
2. 29
3. 31
4. 32
5. 41

Answer:

Approach Solution (1):

The venn diagram doesn’t give you the answer directly. You do have to do the calculations shown above.

Number divisible by 2 = 50
Number divisible by 3 = 33
Number divisible by 5 = 20

Number divisible by 6 (LCM of 2 and 3) = 16
Number divisible by 10 (LCM of 2 and 5) = 10
Number divisible by 15 (LCM of 3 and 5) = 6
Number divisible by 30 (LCM of 2, 3, and 5) = 3

Total = n(A) + n(B) + n(C) – n(A and B) - n(B and C) - n(A and C) + n(A and B and C) = 50 + 33 + 20 – 16 – 10 – 6 + 3 = 74
Number not divisible by 2, 3, or 5 = 100 – 74 = 26

Correct Option: A

Approach Solution (2):

We may also consider that - from 1 to 100 – \(\frac{1}{2}\) of the numbers will be divided by 2 and \(\frac{1}{2}\) will not.
Similarly, from 1 – 100 – \(\frac{1}{3}\)of the numbers will be divided by 5 and \(\frac{4}{5}\) will not.
And, from 1 – 100 – \(\frac{1}{5}\) of the numbers will be divided by 5 and \(\frac{4}{5}\)will not.
Hence, no. of the number which are not divisible by 2, 3, and 5 = \(100*\frac{1}{2}*\frac{2}{3}*\frac{4}{5} = \frac{80}{3} = 26.666\)

Ignoring the decimal since no of number can’t be decimal leaves us with 26.

Correct Option: A

Approach Solution (3):

From 1 to 100, 50 numbers are even, and 50 are odd. therefore, we can eliminate right away 50 numbers.
Now, multiples of 5… let’s see the multiples of 5 that are not odd
5, 15, 25, 35, 45, 55, 65, 75, 85, 95 – 10 numbers – another 10 numbers eliminated
Multiples of 3 that are not multiples of 5 and are not odd
3, 9, 21, 27, 33, 39, 51, 57, 63, 69, 81, 87, 93, 99 – another 14 eliminated
50 – 10 – 14 = 26

Correct Option: A

“How many integers from 1 to 100 are not divisible by 2, 3, and 5?”- 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. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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