The probability that a given positive integer lying between 1 and 100 (both…

2014

The probability that a given positive integer lying between 1 and 100 (both inclusive) is NOT divisible by 2, 3 or 5 is ______ .

Attempted by 60 students.

Show answer & explanation

Correct answer: 0.259 to 0.261

Key idea: Use the inclusion-exclusion principle to count how many integers from 1 to 100 are divisible by 2, 3, or 5, then subtract from 100.

  1. Count multiples of each prime: multiples of 2 = 100/2 = 50; multiples of 3 = 100/3 = 33; multiples of 5 = 100/5 = 20.

  2. Subtract multiples counted twice (pairwise lcms): multiples of lcm(2,3)=6 → 100/6 = 16; lcm(2,5)=10 → 100/10 = 10; lcm(3,5)=15 → 100/15 = 6.

  3. Add back multiples counted three times (lcm of all three): lcm(2,3,5)=30 → 100/30 = 3.

  4. Total numbers divisible by 2, 3, or 5 = 50 + 33 + 20 - (16 + 10 + 6) + 3 = 74.

  5. Numbers NOT divisible by 2, 3, or 5 = 100 - 74 = 26. Probability = 26/100 = 13/50 = 0.26.

Final answer: 13/50 (which is 0.26).

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