Name: pat walsh
Who is asking: Parent
Level: All

Question:
why does it work when you add the digits of a number then divid by three to see if the number is divisible by three

Hi Pat,

Suppose that we have a number such as 834 567 and wonder if it is a multiple of 3.

834 567= 8x105 + 3x104 + 4x103 + 5x102 + 6x101 + 7
 = 8 (99 999 + 1) + 3 (9999 + 1) + 4 (999 + 1) + 5 (99 + 1) + 6 (9 + 1) + 7
 = 8 (99 999) + 3 (9999) + 4 (999) + 5 (99) + 6 (9) + (8 + 3 + 4 + 6 + 7)

Now we know that 3 divides 9 so if we ask whether 3 divides 834 567 or not we can reduce our problem as to whether or not 3 divides (8 + 3 + 4 + 5 + 6 + 7), i.e. does 3 divide the sum of the digits of the original number?

You can see that "3" in the argument above could be replaced by "9" so that an integer is divisible by 9 precisely if the sum of its digits is divisible by 9.

Cheers,
Penny
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