# Life long sharing . . .

### Maths – How many 3-digit numbers are multiple of 21?

Posted on: April 13, 2017

How many 3-digit numbers are multiple of 21?

Yes, we can solve this question by counting, but there is another way to solve this type of problem.

First, find the smallest  3-digit number divisible by 21:

• the smallest 3-digit number is 100
• divide 100 by 21 gives 4.76, which rounds up to 5
• so the smallest 3-digit multiple of 21 will be 5 x 21 = 105.

Next, find out the largest 3-digit number divisible by 21:

• the largest 3-digit number is 999
• divide 999 by 21 gives 47.57, which rounds down to 47
• so the largest 3-digit multiple of 21 will be 47 x 21 = 987.

Method 1: The 3-digit multiples of 21 form the arithmetic sequence

5 x 21, 6 x 21, …, 46 x 21, 47 x 21

Hence, the number of 3-digit numbers which are multiples of 21 is 47 – 5 + 1 = 43

Method 2: Find out how many intervals by 21 are between these two numbers.

(987 – 105) ÷ 21 = 42

Hence, the number of multiples of 21 from 105 to 987 inclusively is 42 + 1 = 43.

Now, let’s try this:

How many 3-digit numbers are multiple of 23?

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