Mathematics, 03.08.2019 10:30 Arellano4672

Why does multiple of 3 add up digits trick work?

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You must be referring to the quick way to determine whether a given integer is divisible by 3; that is, 3 divides an integer

whenever the digits of

sum to a multiple of 3.

Suppose

has $n\ge1$ digits. We can expand it as a sum of the numbers in any given digits place by the corresponding power of 10. For example, if x=2148

, we can write

$2148=2\times10^3+1\times10^2+4\times10^1+8\times10^0$

More generally, if

$x=d_{n-1}d_{n-2}\ldots d_1d_0$

(where d_i

denotes the numeral in the 10^i

-th's place), then we have the expansion

$x=d_{n-1}10^{n-1}+d_{n-2}10^{n-2}+\cdots+d_110^1+d_0$

Notice that for any integer

, we have

$10^k-1=\underbrace{99\ldots99}_{k\text{ 9s}}$

which is clearly divisible by 3. So from each power of 10 in the expansion of

, we can add and subtract 1, then rearrange the terms of the sum:

$x=d_{n-1}10^{n-1}+\cdots+d_110^1+d_0$
$x=d_{n-1}(10^{n-1}-1+1)+\cdots+d_1(10^1-1+1)+d_0$
$x=d_{n-1}(10^{n-1}-1)+\cdots+d_1(10^1-1)+(d_{n-1}+\cdots+d_1+d_0)$

We know 10^k-1

is divisible by 3, which means the remainder upon dividing

by 3 is just the sum of the digits of

. If this remainder is divisible by 3, then so must be the original number,

.

Back to our previous example: if x=2148

, then we have the expansion

$2148=2\times10^3+10^2+4\times10+8$
2148=2(999+1)+(99+1)+4(9+1)+8

$2148=2\times999+99+4\times9+(2+1+4+8)$

Dividing through by 3, we get a remainder of 2+1+4+8=15

, which is divisible by 3, and so 2148 must also be a multiple of 3.

In case for some reason you're not convinced 15 is a multiple of 3, you can apply the same trick:

15=10+5

Dividing through by 3 leaves a remainder of 1+5=6

, which is also a multiple of 3, so that 15 must be, too.

Answer from: Quest

points are handed out whenever you post answers.

brainliest answers are handed out by the person asking the question, so whether your answer is picked depends entirely on whether the asker bothers to pick

if you're asking about your rank, it depends on having a certain number of points and a certain number of brainliest answers. for example, at the time of this writing, i have 44338 points out of 15000 needed for the next (nearly triple the requirement! ) but i only have 24 out of 50 required brainliest answers. this number decays, meaning after a certain amount of time (i think about a month), the total number of brainliest answers you've earned doesn't count toward this total. (i've been stuck at the same rank for a long

the lesson to learn here is that you shouldn't worry about points or rank. chances are, no matter how good an answer you may provide, your efforts will go mostly unnoticed unless you focus on the newest/most active questions that get posted.

Answer from: Quest

the formula of a slope:

$m=\dfrac{y_2-y_1}{x_2-x_1}$

we have the points (-2, 7) and (2, 3). substitute:

$m=\dfrac{3-7}{)}=\dfrac{-4}{2+2}=\dfrac{-4}{4}=-1$

answer: the slope is equal -1.

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