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 Subject: "Back to back pairs" Previous Topic | Next Topic
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Member since Nov-5-04
Nov-05-04, 12:23 PM (EST)   "Back to back pairs"

 I was working on pairs on two digit numbers where their product stays the same when the order of the digits are reversed. For example 23x96=32x69 or 12x63=21x36 or 43x68 = 34x86 or 12x84 = 21x48. Can any one tell me how many such pairs are there & if there is any formula on such pairings???/Adi

Ramsey_KJ
Member since Sep-23-04
Nov-06-04, 07:29 PM (EST)   1. "RE: Back to back pairs"
In response to message #0

 If the pairs are denoted A(1)A(2) and B(1)B(2) then A(1)*B(1) must equal A(2)*B(2). This implies that the if the product A(1)*B(1) = P(1)^X * P(2)^Y * P(3)^Z, then the sum of X,Y and Z must be greater than 2 but the exponent of 1 can only be 1 since higher exponents give only 1. Because of the limitation that A(1), A(2), B(1), and B(2) must all be less than 10, there are only a few possible combinations, e.g. for P(n) > 3 there are no possible solutions and P(n) must equal 1, 2 or 3. It is then easy to determine the few possible combinations of A(1)(A2) and B(1)B(2) {of course you ignore the reversed orders}. I get a total of 14 including the 4 that you gave. Thanks for your problem.Have a Good DayKJ Ramsey

Owen guest
Dec-23-04, 12:30 PM (EST)

2. "RE: Back to back pairs"
In response to message #0

 According to the previous replyif the pairs are denoted A(1)A(2) and B(1)B(2) then A(1)*B(1) must equal A(2)*B(2) where A(1), A(2), B(1), and B(2) can only be positive integer less than 10 (1 to 9). Those integers can produce the ratio 1/2 1/3 2/3 1/4 and 3/4 where A(1)A(2) and B(1)B(2) can be createdratio 1/2 : (1,2) (2,4) (3,6) (4,8)ratio 1/3 : (1,3) (2,6) (3,9)ratio 1/4 : (1,4) (2,8)ratio 2/3 : (2,3) (4,6) (6,9)ratio 3/4 : (3,4) (6,8)Then, we can built a pair of number by selecting 2 members within each ratio groupratio 1/2 : select 2 out of 4 = 4C2 = 6 waysratio 1/3 : select 2 out of 3 = 3C2 = 3 waysratio 1/4 : select 2 out of 2 = 2C2 = 1 wayratio 2/3 : select 2 out of 3 = 3C2 = 3 waysratio 3/4 : select 2 out of 2 = 2C2 = 1 way14 ways in total and they are ...12 4212 6312 8413 6213 9314 8223 6423 9624 6324 8426 9334 8636 8446 96:) lover guest
Mar-22-05, 11:16 AM (EST)

3. "RE: Back to back pairs"
In response to message #2

 this stuff is bull. well i have a teacher that doesn't teach us any thing.she can't speack right

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