The number of positive integers not greater than 100, which are not di

Question

The number of positive integers not greater than 100, which are not divisible by 2, 3 or 5 is

(A) 26 
(B) 27
(C) 31 
(D) 32
(E) 33

Are You Up For the Challenge: 700 Level Questions: 700 Level Questions

sujoykrdatta · Accepted Answer

Least common multiple of 2,3,5 = 30
Multiple of 30 near 100 is 90.

From the 90 numbers, 90/2 = 45 are divisible by 2. Hence, the remaining 45 are NOT divisible by 2.

Of these 45 numbers, 45/3 = 15 are divisible by 3. Hence, the remaining 30 are NOT divisible by 3.

Of these 30 numbers, 30/5 = 6 are divisible by 5. Hence, the remaining 24 are NOT divisible by 5.

Thus, till 90, there are 24 numbers that are not divisible by 2, 3 or 5

From 91 to 100, there are 2 more numbers - 91 and 97

Thus, total numbers = 24 + 2 = 26

Answer A

Posted from my mobile device

flash007 · Answer

There are a total of 25 prime numbers between 1 to 100.
Out of these primes, we remove 2,3 & 5 so we are left with  22  numbers.

There are  3  multiples (7*7,7*11,7*13) of 7 which are less than 100 and not divisible by 2,3 or 5.
also, 1 is not divisible by 2,3, or 5 and is less than 100.

Therefore answer=22+3+1=26
IMO A

ravigupta2912 · Answer

Total numbers from 1 to 100, inclusive = 100
Divisible by 2 = (100-2/2)+1 = 50
Divisible by 3 (but excluding multiples of 6 being the LCM of 2,3) = (99-3/6)+1 = 17
Divisible by 5 (but not divisible by either 2 or 3) = 25, 35, 55, 65, 85,95 = 7

Total divisible by either 2 or 3 or 5 = 74
Total not divisible = 100-74 = 26

Answer - A

ScottTargetTestPrep · Answer

Solution:

The integers satisfy the condition are 1, primes excluding 2, 3 and 5, and all the composite numbers whose smallest prime factor is at least 7.

The number of primes less than 100 is 25; so, excluding 2, 3, and 5, there are 22 primes satisfying the condition.

The composite numbers less than 100 with the smallest prime factor at least 7 are 7^2 = 49, 7 x 11 = 77 and 7 x 13 = 91.

Therefore, including the number 1, there are 1 + 22 + 3 = 26 such numbers.

Answer: A

MathRevolution · Answer

Total number of positive integers not greater than 100= 100

Not divisible by 2, 3, or 5 = Total - Divisible by 2,3, or 5

Total number of positive integers not greater than 100 divisible by 2: 50
Total number of positive integers not greater than 100 divisible by 3: 33
Total number of positive integers not greater than 100 divisible by 5: 20 
 Total number of positive integers not greater than 100 divisible by 2 and 3 both: 16  
Total number of positive integers not greater than 100 divisible by 2 and 5 both: 10  
Total number of positive integers not greater than 100 divisible by 3 and 5 both: 6   
 Total number of positive integers not greater than 100 divisible by 2, 3, and 5: 3

Total number divisible by 2, 3, or 5 =  50 + 33 + 20   - 16 - 10 - 6   + 3  = 74

Therefore, not divisible by 2, 3, or 5 = 100 - 74 = 26

Answer A

AbhishekAnil · Answer

interesting approach, but how do we know that we are not missing any number that is no a prime number.

The number of positive integers not greater than 100, which are not di

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

The number of positive integers not greater than 100, which are not di

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards