2006 UNCO Math Contest II Problems/Problem 2

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Problem

If $a,b$ and $c$ are positive integers, how many integers are strictly between the product $abc$ and $(a+1)(b+1)(c+1)$ ? For example, there are 35 integers strictly between $24=2*3*4$ and $60=3*4*5.$

Solution

We start by expanding $(a + 1)(b + 1)(c + 1)$, and in doing so we obtain $abc + ab + ac + a + bc + b + c + 1$. We then subtract $abc$ from $abc + ab + ac + a + bc + b + c + 1$, to get $ab + ac + a + bc + b + c + 1$. We finally subtract $1$ from this value, as the problem asks for how many integers are strictly in between, and we get our answer of $\boxed{ab + ac + bc + a + b + c}$ as desired.

See Also

2006 UNCO Math Contest II (ProblemsAnswer KeyResources)
Preceded by
Problem 1
Followed by
Problem 3
1 2 3 4 5 6 7 8 9 10
All UNCO Math Contest Problems and Solutions
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