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I have exams on number theory coming up.And this is something I don´t really understand, how to handle such tasks. Could anyone please explain it to me in a very understandable way (just studying primary school teaching..) Thank you folks.

If 7/2a +b, then 7/ 100 a +b.

I could transform it to 100a+b= 7l and 2a+b= 7k But how to prove now? Also the task says " after proving the divisibility, show one rule of divisibility for seven. How does this work? Thank you so much.Looking forward to your answers. Sophia

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If $7\mid x$, then $7\mid y$ if and only if $7|(y-x)$ – egreg Mar 17 '13 at 15:53
Do you seek to understand only why the above inference is true or, more generally, do you seek to understand the genesis of this inference when devising a test for divisibility by $7$? – Math Gems Mar 17 '13 at 17:17
up vote 1 down vote accepted

Short Answer

If $a|b$ then $a|a.n + b$

Long Answer

Known $$ 7|2a +b\tag1$$

Need to prove

$$7| 100 a +b$$ $$\Rightarrow 7| (98+2) a +b$$ $$\Rightarrow 7| 98a+2a +b$$ $$\Rightarrow 7| 98a+7|2a +b$$ $$\Rightarrow 7| 7^2.2.a+7|2a +b$$

As $7| 7^2*2a$ and from $(1)$ we have $7|2a +b$

So $$7| 100 a +b$$

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thank you so much for the answer!! – Sophia Mar 17 '13 at 15:59

It's very easy, have to use only that $7|98$ (as $98=49+49$ and $49=7^2$).

The corresponding rule of divisibility is, in an example: $7\,|\,105 \iff 7\,|\,(2\cdot 1+5)$ or $7,|\,522 \iff 7\,|(2\cdot 5+22)$, ...

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thank you i can see the prove. but what is that rule pf divisibility? could you please comment on that? thank you! – Sophia Mar 17 '13 at 15:58
The rule helps deciding whether a given number is divisible by seven or not. For example for $12544$, this can be written as $125\cdot100+44$, by the statement we have that it's divisible by $7$ iff $125\cdot 2+44$ is so, that is, $294$, by the statement again, it reduces to $7\,\overset?|\,(2\cdot 2+94)$, which now happens to hold.. – Berci Mar 17 '13 at 17:16
okay thank you!! – Sophia Mar 17 '13 at 17:39

Hint $\rm\ \ 100 a\!+\! b\ =\ 2a\!+\!b\: +\: 7\,(14 a)$

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thank you, but why is it 14a?? – Sophia Mar 17 '13 at 16:06
@Sophia Because $\rm\ x=14\:$ solves $\rm 100 = 2 + 7 x.\ $ – Math Gems Mar 17 '13 at 16:26

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