Given equation: 1 + 100x + 102y = xy

To find the integral pairs (x, y) that satisfy the equation, we can rearrange the equation as:

xy - 100x - 102y + 1 = 0

Using the quadratic formula:

The equation can be rewritten in the form of a quadratic equation:

xy - 100x - 102y + 1 = 0

Rearranging the terms:

xy - 100x - 102y = -1

Adding 10100 to both sides:

xy - 100x - 102y + 10100 = 10100 - 1

xy - 100x - 102y + 10100 = 10099

Now, let's add 10100 to both sides:

xy - 100x - 102y + 10100 = 10099

(x - 102)(y - 100) = 10099

Factors of 10099:

To find the factors of 10099, we can prime factorize it:

10099 = 7 * 1443

So, the factors of 10099 are: 1, 7, 1443, and 10099.

Possible values of (x - 102) and (y - 100):

Since (x - 102)(y - 100) = 10099, we need to find the possible combinations of (x - 102) and (y - 100) that result in the factors of 10099.

Possible combinations are:

(x - 102) = 1, (y - 100) = 10099
(x - 102) = 7, (y - 100) = 1443
(x - 102) = 1443, (y - 100) = 7
(x - 102) = 10099, (y - 100) = 1

Now, solving for x and y:

(x - 102) = 1, x = 103
(y - 100) = 10099, y = 10199

(x - 102) = 7, x = 109
(y - 100) = 1443, y = 1543

(x - 102) = 1443, x = 1545
(y - 100) = 7, y = 107

(x - 102) = 10099, x = 10101
(y - 100) = 1, y = 101

So, the integral pairs (x, y) that satisfy the equation are:

(103, 10199)
(109, 1543)
(1545, 107)
(10101, 101)

Therefore, there are 4 integral pairs (x, y) that satisfy the equation.