Math Is Fun Forum

rida

Real Member

Registered: 2006-09-25

Posts: 839

Trick to finding multiples of 3

To find multiples of 3 you need to keep on adding the digits until you have a single digit and if that digit is a 3,6 or 9 the number is a multiple of 3.

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mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Trick to finding multiples of 3

Very true. In addition, if the digit is a 9 then the original number is a multiple of 9.
Does 0 count as a multiple of 3?

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luca-deltodesco

Member

Registered: 2006-05-05

Posts: 1,470

Re: Trick to finding multiples of 3

i dont quite get how tihs works...

13 isnt a multiple of 3, and 19 certainly isnt a multiple of 9...

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JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Trick to finding multiples of 3

i dont quite get how tihs works...

Let’s take an example: 314159265
Add the digits: 3 + 1 + 4 + 1 + 5 + 9 + 2 + 6 + 5 = 36
Add the digits of the resulting sum: 3 + 6 = 9
(You keep adding the digits of the resulting sums until you get a single-digit number.)
9 is a multiple of 3.
∴ 314159265 is a multiple of 3.

Another example: 2718281828
Add the digits: 2 + 7 + 1 + 8 + 2 + 8 + 1 + 8 + 2 + 8 = 47
Add again: 4 + 7 = 11
And again: 1 + 1 = 2
2 is not a multiple of 3.
2718281828 is not a multiple of 3.

And mathsyperson, yes: 0 is a multiple of 3. An integer a is a multiple of an integer b iff a = bc for some integer c; since 0 = 3×0, 0 is thus a multiple of 3.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Trick to finding multiples of 3

13 = 1 + 3 = 4 which is not a multiple of 3, so 13 isn't a multiple of 3.
19 = 1 + 9 = 10 = 1 + 0 = 1.  Same.
12 = 1 + 2 = 3
105 = 1 + 0 + 5 = 6

You can do the same thing with any base.  11 is a good one to do as it turn out to be:

But some can be just weird.  For example, try doing it for base 7.

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lightning

Real Member

Registered: 2007-02-26

Posts: 2,060

Re: Trick to finding multiples of 3

3x0=0
3x1=3
3x2=6
3x3=9
3x4=12
3x5=15
3x6=18
3x7=21
3x8=24
3x9=27
3x10=30

that should be the answer to your question

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mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Trick to finding multiples of 3

And mathsyperson, yes: 0 is a multiple of 3. An integer a is a multiple of an integer b iff a = bc for some integer c; since 0 = 3×0, 0 is thus a multiple of 3.

I thought it was, but if that's the case then rida's definition isn't quite right. As well as 3, 6, or 9, it also works when all the digits add to 0. Of course, that only happens when the original number was 0 anyway.

If I've followed Ricky's method correctly, then the formula that you get out of it would be a horrible thing that involved a loop of 6 different coefficients for each digit. It'd probably be easier to just try to divide whatever your number is by 7.

I know an alternate way of finding whether a number is a multiple of 7 that's possibly slightly easier than Ricky's (although just dividing by 7 would probably still be easier).

Take the last digit of your number, double it and take it away from the rest of the number.
For example, for 323274 we would take the 4, double it and take it away from 32327, making our new number 32319.

Reiterating this gives 3213, then 315, then 21.
21 is a multiple of 7, and so 323274 also is.

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