SOLUTION: What is the smallest positive integer that,
when multiplied by 7, yields a product
in which all of the digits are 5’s?
Algebra.Com
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) (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 then must be a multiple of 7.
We must find a that is a multiple of 7.
, , , and are not,
but so
which shows that 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
is ;
the remainder of dividing by 7
is ;
the remainder of dividing by 7
is ,
which differs by a multiple of 7 from , and so on.
The remainder of dividing by 7
differs by a multiple of 7 from
,
meaning that 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.
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