Remainder on division by 9
Add the digits again and again: the last digit is the remainder.
Remainders of tens
Divide 10 by 9: it leaves remainder 1. Divide 20 by 9: it leaves remainder 2.
What remainder do you expect when 50 is divided by 9?
What this lesson covers
The idea
Each digit is the remainder its place-value part leaves on division by 9, so adding the digits repeatedly until a single digit remains gives the remainder when the number is divided by 9.
Remainders of tens
Divide 10 by 9: it leaves remainder 1. Divide 20 by 9: it leaves remainder 2.
What remainder do you expect when 50 is divided by 9?
- 5
- 0
- 50
Add the digits again
Press Add the digits and look at what each step takes away. If more than one digit is left, press again. Change the number with ▲ ▼ and try more.
The last digit is the remainder
Each digit is the remainder its place-value part leaves on division by 9, so adding the digits repeatedly until a single digit remains gives the remainder when the number is divided by 9.
For 7309: 7 + 3 + 0 + 9 = 19 1 + 9 = 10 1 + 0 = 1
Why? A digit times its place value leaves that digit over: 7 × 1000 = 7 × 999 + 7. The part 7 × 999 is a multiple of 9, and 7 is left.
Each step takes away a multiple of 9, so 7309 is 1 more than a multiple of 9: the remainder is 1.
If the single digit is 9, it is one more whole group of 9. The remainder is 0: the number is divisible by 9.
Notes
Each digit is the remainder its place-value part leaves on division by 9, so adding the digits repeatedly until a single digit remains gives the remainder when the number is divided by 9.
Check yourself
What remainder does 427 leave when divided by 9?
Answer: 4
4 + 2 + 7 = 13 and 1 + 3 = 4. So 427 is 4 more than a multiple of 9 (427 = 9 × 47 + 4). The remainder is 4.
The digits of a number add up to 27, and 2 + 7 = 9. What remainder does it leave when divided by 9?
Find the remainder when 5864 is divided by 9.
Answer: 5
5 + 8 + 6 + 4 = 23 and 2 + 3 = 5. So 5864 leaves remainder 5 (5864 = 9 × 651 + 5).
Why does the digit 3 in 7309 (hundreds place) add just 3 to the remainder?
- 9. A remainder on division by 9 is always less than 9. A 9 left over makes one more group of 9.
- 2. That is the first digit of 27. Add 2 + 7 to get a single digit first.
- 0 — correct. Yes! The single digit 9 is one more whole group of 9, so nothing is left over. The number is divisible by 9.
- 7. That is the second digit of 27. Add 2 + 7 to get a single digit first.
- Because 300 = 3 × 99 + 3: after the multiple of 9, 3 cells are left — correct. Yes! 3 × 99 is a multiple of 9. Each hundred leaves 1, so 3 hundreds leave 3.
- Because 3 is a small digit. Size is not the reason. A digit 9 in the hundreds place would add 9, and 900 = 9 × 100 is a multiple of 9.
- Because the hundreds place is ignored. No place is ignored. Each place value is 1 more than a multiple of 9, so each digit counts once.