Why adding the digits knows anything
Adding digits ignores the columns and the order. Why does adding digits still tell us anything?
- 4321: digits 4, 3, 2, 1 in their columns.
- Split each digit into a nines part and a leftover.
- Nines pile: 3996 + 297 + 18 = 4311.
- Leftovers: 4 + 3 + 2 + 1 = 10. And 4311 + 10 = 4321.
Every column hides a run of nines
Adding digits still gives the leftover. The ignored part is a multiple of 9.
10 is 9 and 1. 100 is 99 and 1. 1000 is 999 and 1. So a 3 in the tens column is worth 30. That is 3 × 9 = 27, plus 3 more. Split every column that way. The number then has two piles. The nines pile holds all the nines. The rest is the leftovers.
9, 99 and 999 all divide by 3 and by 9. So the whole nines pile does too. Only the leftovers can change the answer. The leftovers, added up, are the digit sum.
Watch out!
"It divides by 3 if it ends in 3, 6, or 9." That is not so. Look at 13. It ends in 3. But 13 ÷ 3 leaves 1 over. The test for 3 adds every digit, not the last one.
Slide a number. Watch every column split into nines and a leftover.
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