# Three-digit numbers not divisible by 3

**Problem**

How many three-digit numbers are not divisible by 3?

**Solution**

There are 900 three-digit numbers, namely 100, 101, 102, ..., 999. The first three-digit number that is exactly divisible by 3 is 102 and the last is obviously 999. The numbers 102, 105, 108, ..., 999 form an arithmetic progression with common difference, d = 3.

By Arithmetic progression:

$a_n = a_1 + (n - 1)d$

$999 = 102 + (n - 1)(3)$

$n = 300$

There are 300 three-digit numbers that are divisible by 3 and there are 900 three-digit numbers. Thus, the three-digit numbers that are not divisible by 3 is:

$\text{Required } = 900 - n$

$\text{Required } = 600$ *answer*

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