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October 21, 2014

October 21, 2014

Posted by **Katti** on Wednesday, July 7, 2010 at 2:01am.

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 -
**Reiny**, Wednesday, July 7, 2010 at 7:48amNo, 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 -
**Anonymous**, Thursday, July 8, 2010 at 12:39amReiny, Thanks for helping to clarify and working out the problem.

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