Question

The total number of 3-digit numbers, whose greatest common divisor with 36 is 2, is _______.

Answer 1

Correct Answer: 150

To find the total number of 3-digit numbers whose greatest common divisor (GCD) with 36 is 2, we can follow these steps:

1. Note that a 3-digit number ranges from 100 to 999.
2. The GCD condition implies a number must contain the factor 2, but not 4, 3, 6, 9, 12, or 18, as 36=2²×3², and we want GCD to be exactly 2.
3. Identify numbers divisible by 2 (even numbers) between 100 and 999:
   • Start: 100, End: 998
     Formula for total numbers: (end-start)/step+1
   • Here, step is 2: (998-100)/2+1=450 such numbers.
4. Exclude numbers divisible by 4 (2²), as their GCD with 36 would be 4:
   • Start: 100, End: 996
     Step: 4
     Calculation: (996-100)/4+1=225
5. Exclude numbers divisible by 6 (2×3):
   • Start: 102, End: 996
     Step: 6
     Calculation: (996-102)/6+1=150
6. Next, exclude numbers divisible by 12 (2²×3):
   • Start: 108, End: 996
     Step: 12
     Calculation: (996-108)/12+1=75
7. Apply Inclusion-Exclusion principle:
   • Total = Numbers divisible by 2 - Numbers by 4 - Numbers by 6 + Numbers by 12
   • Total numbers: 450 - 225 - 150 + 75 = 150

The total number of 3-digit numbers with a GCD of 2 with 36 is 150, which falls within the given range (150,150).