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30 September, 02:51

Find all two digit positive integers in which the difference between the integer and the product of its two digits is 12

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  1. 30 September, 06:17
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    Let ab be a 2 digit number, then ab=10a+b, where a and b are digits, a≠0.

    "the difference between the integer and the product of its two digits is 12"

    10a+b - (a*b) = 12

    10a+b-ab=12

    factorize b:

    10a+b (1-a) = 12

    subtract 10 from both sides, to produce a factor (1-a) or (a-1) and then factorize:

    10a-10+b (1-a) = 2

    10 (a-1) + b (1-a) = 2

    (a-1) (10-b) = 2

    (a-1) (10-b) can be (1,2), (2, 1), (-1, - 2), (-2, - 1)

    if a-1=1 and 10-b=2, then a=2, b=8

    if a-1=2, and 10-b=1, then a=3, b=9

    if a-1=-1, then a=0, which is not possible

    if a-1=-2, then a=-1, which is not possible

    so the solutions (a, b) are (2, 8) and (3, 9)

    Then the integers are 28, 39

    Answer: 28, 39
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