For any positive integer n, the length of n is defined as the number of prime factors whose product is n. for example, the length of 75 is 3, since 75 = 3 × 5 × 5. how many two-digit positive integers have length 6 ?

is it zero numbers? 6 is not a prime number, and cannot be included in length values

answer: 204

step-by-step explanation:

we are told that the revenue is

x⋅(4080−10x)=−10x^2+4080x

this is a quadratic function, so to maximize it we should find the x-value of its vertex. recall that the x-value of the vertex of the parabola ax^2+bx+c is given by the formula −b/2a.

so in this case, we have a=−10 and b=4080, so the x-value of the vertex is

−b/2a= −(4080)/2(−10)=204