The first step is to set up 5n/7 = 5/7 + n/7 so to strictly answer your question, except where n is a multiple of 7, then the remainder is n/7. Is this what you're asking about?
Suppose you divide a number N by P, and you get a quotient Q with a remainder R. This can be rewritten in fraction form:
(remember that 0 <= R < P).
In our case, we are dividing n by 7. You get a quotient q with a remainder of 2, or:
Now I want to know what 5n is. Multiply both sides of the equation above by 35 and you'll get
Divide 5n by 7:
The quotient would be 5q + 1, and the remainder would be 3.
This is where the comes from
That would only work for numbers up to ... for instance if you substract from , you get , which isn't exactly the remainder of divided by , while as expected. Then you could argue that repeatedly substracting is a way to do it, true, but dividing is then more efficient.Also, couldn't a shorter way by just looking at 10 mod 7 just be subtracting 7 from 10?
A slightly different approach . . .
When the positive integer is divided by 7, the remainder is 2.
What is the remainder when is divided by 7?
"When is divided by 7, the remainder is 2."
. . Hence: .
. . and: .
Therefore, the remainder is 3.