SOLUTION: What is the smallest positive integer that, when multiplied by 7, yields a product in which all of the digits are 5’s?

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Question 854623: What is the smallest positive integer that,
when multiplied by 7, yields a product
in which all of the digits are 5’s?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
I see that as a problem involving modular arithmetic, but I am not sure of what language/jargon to use, so I will try to use terms we both understand the same way.

If %2255...5%22=%2211...1%22%2A5=7n then %2211...1%22 must be a multiple of 7.
We must find a %2211...1%22 that is a multiple of 7.
1 , 11 , 111 , 1111 and 11111 are not,
but %22111%2C111%22=7%2A15873 so

which shows that highlight%2879365%29 is the smallest positive integer that,
when multiplied by 7, yields a product in which all of the digits are 5’s.

NOTE 1: The number is easy to find with a calculator, but is is easy enough without one.
Without a calculator, you can figure out the remainders of dividing by 7
1, 10, 100, 1000, and so on.
The first ones are 1, 3, 2, 6, 4, and 5.
So the remainder of dividing by 7
1%2B10=11 is 1%2B3=4 ;
the remainder of dividing by 7
1%2B10%2B100=111 is 1%2B3%2B2=6 ;
the remainder of dividing by 7
1%2B10%2B100%2B1000=1111 is 5 ,
which differs by a multiple of 7 from 1%2B3%2B2%2B6=12 , and so on.
The remainder of dividing by 7
%22111%2C111%22=%22100%2C000%22%2B%2210%2C000%22%2B%221%2C000%22%2B1000%2B10%2B1
differs by a multiple of 7 from
5%2B4%2B6%2B2%2B3%2B1=21=7%2A3 ,
meaning that %22111%2C111%22 is a multiple of 7.

NOTE 2:
This is the kind of question I would ask in a forum at the artofproblemsolving website.
There you are likely to find a middle school student who could answer the question.