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

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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

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

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
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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 [LCM of 2 and 3 is 6]
Total number of positive integers not greater than 100 divisible by 2 and 5 both: 10 [LCM of 2 and 5 is 10]
Total number of positive integers not greater than 100 divisible by 3 and 5 both: 6 [LCM of 3 and 5 is 15]
Total number of positive integers not greater than 100 divisible by 2, 3, and 5: 3 [LCM of 2,3, and 5 is 30]

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
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