Number System Question 6 - CAT 2020

Step-by-Step Explanation

To find how many numbers from 1 to 120 are not divisible by 2, 5, or 7, we can use the Principle of Inclusion-Exclusion (PIE) to find how many numbers are divisible by them first.

• Multiples of single prime numbers:
  • Divisible by 2: ⌊ (120)/(2) ⌋ = 60
  • Divisible by 5: ⌊ (120)/(5) ⌋ = 24
  • Divisible by 7: ⌊ (120)/(7) ⌋ = 17
• Multiples of two-prime combinations (double counting overlaps):
  • Divisible by 2 and 5 (multiples of 10): ⌊ (120)/(10) ⌋ = 12
  • Divisible by 2 and 7 (multiples of 14): ⌊ (120)/(14) ⌋ = 8
  • Divisible by 5 and 7 (multiples of 35): ⌊ (120)/(35) ⌋ = 3
• Multiples of all three primes (triple counting overlap):
  • Divisible by 2, 5, and 7 (multiples of 70): ⌊ (120)/(70) ⌋ = 1

Now, calculate the total number of integers divisible by at least one of these primes:

Divisible Pool = (60 + 24 + 17) - (12 + 8 + 3) + 1 = 101 - 23 + 1 = 79

Integers divisible by none = 120 - 79 = 41

Correct Option: B

💡 FundaMakers Tip:

Use the standard set formula Sum(Singles) - Sum(Doubles) + Triple to handle overlapping multiples accurately. This systematic process prevents you from accidentally double-counting numbers like 10 or 70.