Hello i'm acually a student who needs help with math. My question is:

Why is it that when you add the digits of a number you can tell what the multiples of that number are.

Example: 12131313111,

1+2+1+2+1+3+1+1+1=18,

therefore 12131313111 is divisble by 2, 9, 18, & 3 because those numbers are divisble by 18.

Hi,

The rule for checking divisibility by adding the digits is something that works for 3 and 9 but not for 2. A number is divisible by 2 precisely when the last digit of the number is divisible by 2. Thus 376 and 2796520 are divible by 2 since 6 and 0 are divisible by 2, but 2975 and your number 2131313111 are not since 5 and 1 are not divisible by 2.

To see why this works for 9 look at look at an example, say 2427.

2427
= 2x1000 + 4x100 + 2x10 + 7
= 2x(999+1) + 4x(99+1) + 2x(9+1) + 7
= 2x999 + 2x1 + 4x99 + 4x1 + 2x9 + 2x1 + 7
= 2x999 + 4x99 + 2x9 + (2 + 4 + 2 + 7)
= 9(2x111 + 4x11 + 2x1) + (2 + 4 + 2 + 7)
Thus

2427 = 9(2x111 + 4x11 + 2x1) + (2 + 4 + 2 + 7)     *

On the right side, 9 divides 9(2x111 + 4x11 + 2x1) and thus if 9 divides 2 + 4 + 2 + 7, the sum of the digits, then 9 divides 2427.

The equation marked * can be rewritten

2 + 4 + 2 + 7 = 2427 - 9(2x111 + 4x11 + 2x1)

and hence if 9 divides 2427 then 9 divides 2 + 4 + 2 + 7

In this example 9 does not divide 2 + 4 + 2 + 7 = 15 so 9 does not divide 2427.

For divisibility by 3 look at the equation marked * again.

On the right side, 3 divides 9(2x111 + 4x11 + 2x1) and thus if 3 divides 2 + 4 + 2 + 7, the sum of the digits, if and only is 3 divides 2427.

Here 3 does divide 2 + 4 + 2 + 7 = 15 and hence 3 divides 2427.

I hope this helps,
Penny
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