Algebra

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Need help plz. Don't really get this "reversing the digits" What is this really called in math language?

Prob:Find the product of 32 and 46. Now reverse the digits and find the product of 23 an 64. The products are the same. Does this happen with any pair of two digit numbers?
Find Two other pairs of two-digit numbers(with different digits)that have this property.
Explain your thinking in solving this problem and also include any mathematical work.
TY

  • Algebra -

    No, it does not happen with any pair of 2-digit numbers, all we need is one counterexample,
    e.g. 25 and 13
    25x13 = 325
    52x31 = 1612 , so not generally true

    When is it true?
    for the first number, let the tens digit be a, and the unit digit be b, thus the number is 10a + b
    for the 2nd number, let the tens digit be c, and the unit digit be d, thus the number is 10c + d

    product of pair
    = (10a+b)(10c+d) = 100ac+10ad+10bc+bd
    product of numbers reversed
    = (10b+a)(10d+c) = 100bd+10bc+10ad+ac

    we want
    100ac+10ad+10bc+bd = 100bd+10bc+10ad+ac
    which gives
    99ac = 99bd
    ac = bd

    which shows that it must be true if the product of their tens digit equals the product of their units digits.
    check for given pair
    32 and 46
    product of tens digit = 3x4 = 12
    product of unit digits = 2x6 = 12

    another pair would be
    82 and 14

    I will let you find a second example

  • Algebra -

    Reiny, Thanks for helping to clarify and working out the problem.

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