Proof for divisibility by $7$ One very classic story about divisibility is something like this.

A number is divisible by $2^n$ if the last $n$-digit of the number is divisible by $2^n$.
  A number is divisible by 3 (resp., by 9) if the sum of its digit is divisible by 3 (resp., by 9).
  A number $\overline{a_1a_2\ldots a_n}$ is divisible by 7 if $\overline{a_1a_2\ldots a_{n-1}} - 2\times a_n$ is divisible by 7 too.

The first two statements are very well known and quite easy to prove. However I could not find the way on proving the third statement.
PS: $\overline{a_1a_2\ldots a_n}$ means the digits of the number itself, not to be confused with multiplication of number.
 A: Let $x$ be the number you gave, with the full $n$ digits, and let $y$ be the number whose decimal representation is obtained by removing the last digit, namely $a_n$, of $x$.
The following equation is clear:
$$x=10y +a_n.$$
Note that $10y +a_n$ is divisible by $7$ iff $20y+2a_n$ is divisible by $7$.
But $20y+2a_n$ is divisible by $7$ iff $-y+2a_n$ is divisible by $7$  (I subtracted $21y$), and this is the case if and only if $y-2a_n$ is divisible by $7$.  That's exactly what we wanted to show.
We can write the above stuff using the language of congruences if we wish.
About iff: I slipped into jargon. The word (?) "iff" abbreviates "if and only if."
A: HINT $\rm\qquad 7\ |\ 10\ y + x\ \iff\ 7\ |\ y-2\ x\ \ \:$ since $\rm\:\ -2\ (10\ y + x)\ \equiv\ y - 2\ x\ \ (mod\ 7)$
i.e. lines $\rm\ -10\ y = x\ $ and $\rm\ y = 2\ x\ $ are equivalent over $\rm\:\mathbb Z/7\:,$ differing by a unit scaling (of $2$).
A: Regarding divisibility by 7 I created an algorithm that must be applied repetitively to each period of a large number. If the last sum results in a multiple of  7 then the tested number is also a multiple of 7, otherwise it is not. The sum of each period must be added to the sum of the next period.
N = abc
algorithm: – ( (x) + a + b + c + (y) ) mod 7
(x) must be mentally inserted before the hundreds and (y) must be mentally inserted after the ones in such a way that 7 divides both (x)a and c(y). In this rule no digits are inserted before or after the tens.
Numerical examples:
1) N = 462; – ( (1) + 4 + 6 + 2 + (1) ) mod 7 ≡ Ø
2) N = 863; – ( (2) + 8 + 6 + 3 + (5) ) mod 7 ≡ 4
3) N = 1.554; – ( 1 + (4) ) mod 7 ≡ 2; – ( 2 + (3) + 5 + 5 + 4 + (2) ) mod 7 ≡ Ø
4) N = 68.318; – ( 6 + 8 + (4) ) mod 7 ≡ 3; – ( 3 + (6) + 3 + 1 + 8 +(4) ) mod 7 ≡ 3
5) N = 852.655; – ( (2) + 8 + 5 + 2 + (1) ) mod 7 ≡ 3; – ( 3 + (5) + 6 + 5 + 5 + (6) ) mod 7 ≡ 5
To determine the remainder take the digit that forms with the last sum a multiple of 7.
Remainders of the second, fourth and fifth examples:
2) Last sum is 4 → 4(2); 2 is the remainder
4) Last sum is 3 → 3(5); 5 is the remainder
5) Last sum is 5 → 5(6); 6 is the remainder
The application of this rule must be performed in a dynamic way. It is not necessary to write the inserted numbers; all you need is to add the written numbers and the mentally inserted numbers at sight, determine the difference of each sum and its next multiple of 7 (inverse additive) and go ahead till the value of the last inserted digit is added.
The bureaucratic application of the algorithm certainly will not be as quick as its dynamic application.
The rule works because before the use of the inverse additive the values of the digits of the first three digits are multiplied respectively by 3, 1 and 5; after the inverse additive the multipliers change to 4, 6 and 2. Each group of 6 digits of a number are multiplied by 4,6,2,3,1 and 5. These are the remainders determined by Pascal’s criterion regarding divisibility by 7.
In my blogspot I explain how easy is to create new real rules for divisibility by 7 and other aspects of my research. This is the first real rule for divisibility by 7 created by me.
English is not my native language and I am not a Mathematician.
A: $$5(\overline{a_1a_2\ldots a_n})=50(\overline{a_1a_2\ldots a_{n-1}})+5a_n=\overline{a_1a_2\ldots a_{n-1}}-2a_n\pmod{7}$$
A: I'm taking the title of the question to allow general tests for divisibility by 7 even though the specifics of the original question is about a particular test. Apologies if this is not right - I'm still fairly new to mse.
[Source for this answer is The Lore of Large Numbers, by P. Davis, p. 87-88]
A number is divisible by 7 if and only if:
(3 * units' digit) + (2 * tens' digit) - (1 * hundreds' digit) - (3 * thousands' digit) - (2 * ten thousands' digit) + (1 * hundred thousands' digit) is divisible by 7.
If there are more digits present, the sequence of multipliers 3, 2, -1, -3, -2, 1 is repeated as often as necessary. If fewer digits are present, stop when get to the last multiplier.
Example:
$$7 * 457404 = 3201828$$
$$
\begin{array}
&3 & * & 8 & = & 24 \\
2 & * & 2 & = &  4 \\
-1 & * & 8 & = &-8 \\
-3 & * & 1 & = &-3 \\
-2 & * & 0 & = &0  \\
1  & * & 2 & = &2  \\
3 & * & 3 & =  & 9 \\
\end{array}
$$
Since $24 + 4 - 8 - 3 + 0 + 2 + 9 = 28$ and 7 divides 28, $3201828$ is divisible by 7.
Proof for 3 digit numbers
$$N = 100a + 10b + c,$$ then if $S$ is the sum required by the test,
$$S = -a + 2b + 3c.$$
Then $2S = -2a + 4b + 6c$ and
$N + 2S = 98a + 14b + 7c = 7(14a + 2b + c)$.
The sum $N + 2S$ is therefore a multiple of $7$, say $7M$.
Now if $N$ is a multiple of $7$, say $7P$, then $2S = 7M - 7P = 7(M - P).$ Since therefore $2S$ is divisible by $7$ and $S$ is whole, $S$ also is divisible by $7$ (or $S = \frac{7X}{2},$ where $X$ is even because $S$ is whole).
Conversely, if $S$ is a multiple of $7$, say $7Q$, then
$$N = 7M - 14Q = 7(M - 2Q).$$
This means that $N$  must be a multiple of $7$.
Yet another way to do it is to use a similar alternating sum test as for divisibility by 11, but in 3 digit groups, subtracting first, with the sum's divisibility by 7 determining the original number's divisibility by 7.
In the $3201828$ example, this is $828 - 201 + 3 = 630,$ so since $630$ is divisible by $7$, so is $3201828$. I got this from Link.
A: The reason why this formula works is because $21$ is a multiple of $7$ which ends in $1$. Thus, $21a_n$ is always multiple of $7$.
Then
$$7| \overline{a_1...a_n} \Leftrightarrow 7| \overline{a_1...a_n}-21 a_n  \Leftrightarrow 7| \overline{a_1...a_{n-1}0}-20 a_n \Leftrightarrow 7| 10 \cdot[ \overline{a_1...a_{n-1}}-2 a_n] \,.$$
Since $gcd(7, 10)=1$ you get the desired result.
Comment 1 Same way you can get a criteria of divisibility for any number $m$ so that $gcd(m,10)=1$. If this is the case, $m$ has a multiple of the form $10k+1$ and then, you can prove that 
$$m| \overline{a_1...a_n} \Leftrightarrow m|  \overline{a_1...a_{n-1}}-ka_n$$
Comment 2 You can instead look for multiples of $7$ (or $m$)which end in $9$ and add instead of subtraction. 
Since $7|49$ we get that 
$7| \overline{a_1...a_n} \Leftrightarrow 7| \overline{a_1...a_n}+49 a_n  \Leftrightarrow 7| \overline{a_1...a_{n-1}0}+50 a_n \Leftrightarrow 7| 10 \cdot[ \overline{a_1...a_{n-1}}+5 a_n] \,.$$
