Add the digits to test 3 and 6
Show on a big screenA teacher has 4,581 sweets for 3 groups. Can the sweets be shared equally?
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- 4,581 = 4 × 1000 + 5 × 100 + 8 × 10 + 1.
- 1000 = 999 + 1. 100 = 99 + 1. 10 = 9 + 1.
- 999, 99 and 9 are multiples of 3. They leave no remainder.
- The digits are left: 4 + 5 + 8 + 1 = 18. It divides by 3.
The rule
Yes. 4,581 ÷ 3 = 1,527, so each group gets 1,527 sweets.
The digit sum is the digits added up. 4,581 has digit sum 18.
Each place value is a run of nines plus 1. For example, 10 = 9 + 1 and 100 = 99 + 1. A run of nines is a multiple of 3. So each number is its digit sum plus a multiple of 3. Both leave the same remainder when divided by 3.
Rule: A number divides by 3 when its digit sum does. A number divides by 6 when 2 and 3 both divide it.
| Number | Digit sum | By 3? | Even? | By 6? |
|---|---|---|---|---|
| 4,581 | 18 | yes | no | no |
| 2,946 | 21 | yes | yes | yes |
| 7,310 | 11 | no | yes | no |
| 8,305 | 16 | no | no | no |
7,310 has digit sum 11. 11 leaves remainder 2 when divided by 3. So 7,310 also leaves remainder 2.
Watch out!
The last digit does not decide 3. 13 ends in 3 and is not a multiple of 3. 33 is.
Try the lab to finish this lesson.