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What is the number of integral pairs (x, y) that satisfy the equation, 1 + 100x + 102y = xy?
    Correct answer is '6'. Can you explain this answer?
    Verified Answer
    What is the number of integral pairs (x, y) that satisfy the equation,...
    Consider, 1 + 10Ox + 102 y = xy
    xy - 100x - 102y = 1 ...(i)
    Now, (x - 102) (y - 100) = (xy - 100x - 102y) + 102 x 100
    (x - 102)(y - 100) = 1 + 102 x 100 ...[From (i)]
    (x - 102)(y - 100) = 1 +(101 + 1) x (101 - 1)
    = 1 + 1012 - 1 
    = 1012
    Since 101 is a prime number,
    The possible values for {(x - 102), (y-100)} are{1, 1012}, {-1, -1012}, {101, 101}, {- 101, - 101} , {1012, 1} and {-1012, - 1} .
    There are six integral values of {(x - 102), (y -100)} satisfying (i).
    There are six integral values of (x, y) satisfying (i).
    Answer: 6
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    Most Upvoted Answer
    What is the number of integral pairs (x, y) that satisfy the equation,...
    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.
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    What is the number of integral pairs (x, y) that satisfy the equation, 1 + 100x + 102y = xy?Correct answer is '6'. Can you explain this answer?
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