Name: Kelera
Who is asking: Student
Level: Middle

If the sum of the digits of a number is divisible by 9, then the number itself it divisible by 9. Why is that? How do you explain this?

Hi Kelera,

I think that this is easiest to explain using an example. Suppose that the number is 5634. Then

5634 = 5(1000) + 6(100) + 3(10) + 4
  = 5(999 + 1) + 6(99 + 1) + 3(9 + 1) + 4
  = 5(999) + 5 + 6(99) + 6 + 3(9) + 3 + 4
  = 5(999) + 6(99)+ 3(9) + 5 + 6 + 3 + 4
  = (a multiple of 9) + 5 + 6 + 3 + 4

Thus if 5634 is divisible by 9 then, since (a multiple of 9) is divisible by 9 so is 5 + 6 + 3 + 4

Conversely if 5 + 6 + 3 + 4 is divisible by 9 then, since (a multiple of 9) is divisible by 9 so is 5634.

Cheers,
Penny
Go to Math Central