There are 37 positive integers less than or equal to 60 that are divisible by 3, 4, or 5.
Here's how we can determine this using the Principle of Inclusion-Exclusion:
- Divisible by 3: There are 60 / 3 = 20 numbers divisible by 3.
- Divisible by 4: There are 60 / 4 = 15 numbers divisible by 4.
- Divisible by 5: There are 60 / 5 = 12 numbers divisible by 5.
Now, we need to subtract the numbers divisible by more than one of these numbers to avoid overcounting.
- Divisible by both 3 and 4 (i.e., divisible by 12): There are 60 / 12 = 5 numbers.
- Divisible by both 3 and 5 (i.e., divisible by 15): There are 60 / 15 = 4 numbers.
- Divisible by both 4 and 5 (i.e., divisible by 20): There are 60 / 20 = 3 numbers.
Finally, we need to add back the numbers divisible by all three to correct for over-subtracting.
- Divisible by 3, 4, and 5 (i.e., divisible by 60): There is 60 / 60 = 1 number.
Applying the Principle of Inclusion-Exclusion:
Total = (Divisible by 3) + (Divisible by 4) + (Divisible by 5) - (Divisible by 3 and 4) - (Divisible by 3 and 5) - (Divisible by 4 and 5) + (Divisible by 3, 4, and 5)
Total = 20 + 15 + 12 - 5 - 4 - 3 + 1 = 36
Therefore, there are 36 numbers between 1 and 60 (inclusive) that are divisible by 3, 4, or 5. However, because the calculation rounds down, this requires closer attention to ensure no error:
Numbers divisible by 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60 (20)
Numbers divisible by 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60 (15)
Numbers divisible by 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60 (12)
Divisible by 3 and 4: 12, 24, 36, 48, 60 (5)
Divisible by 3 and 5: 15, 30, 45, 60 (4)
Divisible by 4 and 5: 20, 40, 60 (3)
Divisible by 3, 4, and 5: 60 (1)
Therefore: 20 + 15 + 12 - 5 - 4 - 3 + 1 = 36.
The initial answer of 37 appears to be an arithmetic error. The correct answer is 36.
