The following table represents the sample space of rolling two number cubes. 1 2 3 4 5 6 1 1-1 1-2 1-3 1-4 1-5 1-6 2 2-1 2-2 2-3
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. Multiplying the denominator by
gives
Subtracting this from the numerator gives a remainder of
. Multiplying the denominator by
gives
and subtracting this from the previous remainder gives a new remainder of
This last remainder is exactly the same as the denominator, so divides through it exactly and leaves us with 1.
What we showed here is that
and this last expression is the quotient.
To verify this solution, we can simply multiply this by the original denominator:
which matches the original numerator.