Posted by Katti on Wednesday, July 7, 2010 at 2:01am.
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
Reiny, Thanks for helping to clarify and working out the problem.
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