There are 542 positive integers less than 1000 that are multiples of 3, 5, or 7.
How to solve
To find the number of positive integers less than 1000 that are multiples of 3, 5, or 7, we can use the principle of inclusion-exclusion.
First, determine the count of integers divisible by each individual number:
Integers divisible by 3:
999/3= 333
Integers divisible by 5:
999/5 = 199
Integers divisible by 7:
999/7= 142
Next, calculate the count of integers divisible by the pairwise intersections:
Integers divisible by both 3 and 5 (i.e., multiples of 15):
999/15 =66
Integers divisible by both 3 and 7 (i.e., multiples of 21):
999/21 = 47
Integers divisible by both 5 and 7 (i.e., multiples of 35):
999/35= 28
Then, consider the count of integers divisible by all three (i.e., multiples of 3, 5, and 7):
Integers divisible by 3, 5, and 7 (i.e., multiples of 105):
999/105 = 9
Apply the inclusion-exclusion principle to calculate the total count of positive integers that are multiples of 3, 5, or 7:
333+199+142−(66+47+28)+9=542
Therefore, there are 542 positive integers less than 1000 that are multiples of 3, 5, or 7.