Without a calculator: Is 3 a factor of 714?

Can you do this without a calculator? I sometimes have to remind my college students about this "trick". First, here's a terminology refresher. I will use the numbers 2 and 10 to explain:

2 is a factor of 10.
10 is a multiple of 2.
10 is divisible by 2.

Divisibility Rules
2 is a factor if the number is even (number ends in 0, 2, 4, 6 or 8)
3 is a factor if the sum of the number's digits are divisible by 3 (sum means you are adding; I'll explain).
5 is a factor if the number ends in 0 or 5.
6 is a factor if the number is even AND is divisible by 3.
9 is a factor if the sum of the number's digits are divisible by 9 (I'll explain).
10 is a factor if the number ends in 0.


So, back to the original question: Is 3 a factor of 714? According to the rule above, if we add up its digits:

7 + 1 + 4 = 12

Since 12 is divisible by 3, then so is 714.

Another question: Is 6 a factor of 714? How about 741? 
Answer: 2 and 3 are factors are 714 since the number passes "the 3 test" and since the number is even. But is 741 divisible by 6? Does this number pass "the 3 test"? Is it even? (answer below)*

Now, take a look at the multiples of 9: 9, 18, 27, 36, 45, 54, 63, ... If you add up the digits in each of those multiples:
1 + 8
2 + 7
3 + 6
4 + 5

Don't you get 9? So, is 9 a factor of 714? No it isn't since 7 + 1 + 4 = 12 and 9 doesn't divide into 12 evenly.

Is 9 a factor of the HUGE number 702,121,050? Well, find the sum of the digits:

7 + 0 + 2 + 1 + 2 + 1 + 0 + 5 + 0 = 

Now, we can use the Identity Property of addition and not include those zeros in the addition, right?

7 + 2 + 1 + 2 + 1 + 5 = 18

Since 18 is divisible by 9, so is the original number 702,121,050. Hmmm... I wonder if this stuff is the secret behind the Magic Gopher...

Here's some practice from the web along with answers.

(This reminds me that we may need a post about those awesome Properties of Real Numbers... and what are Real Numbers? Oh, so much to discuss!)



* 741 is not divisible by 6 (so 6 is not a factor of 741) since 741 is an odd number.

Revised 9/13/11