Given that a*b + c = 160, where each of a, b and c are positive integers.

Determine the minimum value of a + b*c.

Extra Challenge: Solve this using only pen and paper.

If a*b+c=160 then c=160-ab
and we wish to maximize a+b*c=a+b(160-ab)=-ab^2+160b+a
which is a quadratic in b.
Its maximum occurs when b = -160/(2*-a) = 80/a

Substituting these expression for b and c into the expression to be maximized:
a + 80/a * (160 - a(80/a)) = a + 80/a * 80 = a + 6400/a

By inspection the expression is large if either a is small or a is very large.  If a is large b is not an integer.

If a=1 we get b=80, c=80 and the expression evaluates to 6401

Posted by Jer on 2013-09-17 10:21:15