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# How many positive three digit integers are divisible by neither 2 nor 3?

There are 300 three digit numbers that are divisible by neither 2 nor 3.

There are 999 - 100 + 1 = 900 three digit numbers.

100 ÷ 2 = 50 → first three digit number divisible by 2 is 50 × 2 =100

999 ÷ 2 = 499 r 1 → last three digit number divisible by 2 is 499 × 2 = 998

→ there are 499 - 50 + 1 = 450 three digit numbers divisible by 2.

100 ÷ 3 = 33 r 1 → first three digit number divisible by 3 is 34 × 3 = 102

999 ÷ 3 = 333 → last three digit number divisible by 3 is 333 × 3 = 999

→ there are 333 - 34 + 1 = 300 three digit numbers divisible by 3.

The lowest common multiple of 2 and 3 is 6, so these have been counted in both those divisible by 2 and those divisible by 3.

100 ÷ 6 = 16 r 4 → first three digit number divisible by 6 is 17 × 6 = 102

999 ÷ 6 = 166 r 3 → last three digit number divisible by 6 is 166 × 6 = 996

→ there are 166 - 17 + 1 = 150

→ there are 450 + 300 - 150 = 600 three digit numbers that are divisible by either 2 or 3 (or both).

→ there are 900 - 600 = 300 three digit numbers that are divisible by neither 2 nor 3.

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