The product of two positive integers p and q is 100. What is the large

1.	The product of two positive integers p and q is 100. What is the largest possible value of p+q?
Answer» PQ = 100 ==> P = 100/Q.S = P + Q = 100/Q + Q. By differentiating:dS/dQ = 1 - 100/Q^2. Then, setting dS/dQ = 0 gives:1 - 100/Q^2 = 0Q^2 - 100 = 0 Q^2 = 100 Q = 10.(Pick Q > 0 as we require positive integers) Then, since 1 <= P, Q <= 100, we see that the critical values are:(P, Q) = (10, 10), (1, 100), and (100, 1). We see that the maximum sum occurs when (P, Q) = (1, 100) or (100, 1), which gives 101.

The product of two positive integers p and q is 100. What is the largest possible value of p+q?

Answer»

PQ = 100 ==> P = 100/Q.S = P + Q = 100/Q + Q.

By differentiating:dS/dQ = 1 - 100/Q^2.

Then, setting dS/dQ = 0 gives:1 - 100/Q^2 = 0Q^2 - 100 = 0 Q^2 = 100 Q = 10.(Pick Q > 0 as we require positive integers)

Then, since 1 <= P, Q <= 100, we see that the critical values are:(P, Q) = (10, 10), (1, 100), and (100, 1).

We see that the maximum sum occurs when (P, Q) = (1, 100) or (100, 1), which gives 101.