The product of two positive integers plus their sum is 95. The integers are relatively prime, and each is less than 20. What is the sum of the two integers?

xy + x+y = 95

y(x+1) = 95-x
y = (95-x)/(x+1)

form ordered pairs with positive integer values of x and y

1 47 , y is too big
2 31 too big
3 23
4 * --- * = not an integer
5 15 yeahhhh
6 *
7 11 yeahhh
8 *
9 *

So far I found 2 such pairs
Take their sum in each case.

check for 7,11
product is 77 , sum = 18
What is 77 + 18?

Can you find some more ?

To find the sum of the two integers, we can set up a system of equations based on the given information.

Let's call the two integers x and y.

From the information given, we have two equations:

1. The product of the two integers plus their sum is 95:
xy + (x + y) = 95

2. The two integers are relatively prime. This means that their greatest common divisor (GCD) is 1. In other words, there is no positive integer greater than 1 that divides both x and y evenly.

Now let's solve the system of equations step by step.

First, let's simplify equation 1:
xy + x + y = 95

Next, let's rearrange this equation:
xy + x + y - 95 = 0

Now, we can try to factor this equation:
(xy + x + y) - 95 = 0

Let's use a technique called "completing the square" to factor the equation further:
(x + 1)(y + 1) - 96 = 0

At this point, we need to find two positive integers x and y such that (x + 1)(y + 1) - 96 = 0.

One way to do this systematically is to find the factors of 96 and check if they satisfy the given conditions:

- The factors of 96 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96.

Now, let's substitute these values into the equation (x + 1)(y + 1) - 96 = 0 and check if the integers are relatively prime and each is less than 20.

By trying different pairs, we find that the factors x = 15 and y = 5 satisfy the conditions. (Note: We can switch the values of x and y, as the order does not matter in multiplication.)

So, the two integers are 15 and 5. And the sum of these two integers is:

15 + 5 = 20.

Therefore, the sum of the two integers is 20.