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Since dividing N by 5 gives a remainder of 3, this means that
where 'q' is the quotient (which is some whole number). Multiplying everything by 5 gets us
. If we divide 7N by 5, we then get
. In other words,
. The remainder is
, but it's too big which means that we need to break it up. So
Since the fractional portion is
, this tells us that the remainder is 1.
So the remainder of dividing 7N by 5 is 1.
Note: Try some examples to give you an idea that it works.