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 Subject: "Converting Base Numbers" Previous Topic | Next Topic
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james3760
Member since Mar-24-07
Mar-24-07, 08:06 PM (EST)   "Converting Base Numbers"

 I'm lost.I'm trying to reverse convert two numbers to figure out what the base conversions are and frankly I'm lost.I have two numbers: 37861 and 42050822. A formula was run to convert the first to the second. That's what I'm trying to do.I've run the numbers through dec/hex/octal etc. In fact, here is what I have in converting the first number.Dec 37861Octal 111745TERNARY 1220221021QUINTAL 2202421BIN 1001001111100100Hex 93E5Base - 1 Base - 2 1001001111100100Base - 3 12 2022 1021Base - 4 2103 3211Base - 5 220 2421Base - 6 45 1141Base - 7 21 5245Base - 8 11 1745Base - 9 5 6837Base - 10 37861Base - 11 2 649ABase - 12 1 9AB1Base - 13 1 4305Base - 14 DB25Base - 15 B341Base - 16 93E5Base - 17 7C02Base - 18 68F7Base - 19 59GDBase - 20 4ED1Base - 21 41HJBase - 22 3C4IBase - 23 32D3Base - 24 2HHDBase - 25 2AEBBase - 26 2405Base - 27 1OP7Base - 28 1F85Base - 29 1G0GBase - 30 1C21Base - 31 18CABase - 32 14V5Base - 33 11PABase - 34 WPJBase - 35 UVQBase - 36 T7PMy question is, does any know if there is a formula to run through this and figure out how one number became the other.

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alexb
Charter Member
1980 posts
Mar-24-07, 08:26 PM (EST)    1. "RE: Converting Base Numbers"
In response to message #0

 >I have two numbers: 37861 and 42050822. A formula was run >to convert the first to the second. That's what I'm trying >to do. >Dec 37861 The second number contains an 8 which says its base is at least 9. Since it is longer, its base must be smaller than that of the first number. If the latter is in base 10, then in base 9 it equals 56837 which is much smaller than the second number.What do we learn? The base of the first number is greater than 10 and that of the second is at least 10.Now, since (37861)36 = 537602510 which is still less than the second number, the base of the first number must be greater than 36!

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Member since May-22-05
Mar-29-07, 10:27 AM (EST)    2. "RE: Converting Base Numbers"
In response to message #1

 I think some progress can be made by looking at the divisbility properties of the bases. Since the two digit patters represent the same actual number N, they must be equivalent modulo anything you choose. Let the base of 37861 be A and the base of 42050822 be B.Before considering reductions modulo particular numbers, let's observe that A and B must be relatively prime, because if A and B had any common factor F, then reducing both sides mod F would give N = 1 mod F and N = 2 mod F, so that 1 = 2 mod F, which is only possible for F=1, a case that is excluded from factorizations anyway.Consider mod 2. If A is even, then 37861(A) = 1 mod 2, and if A is odd, then 37861(A) = 1 mod 2 again. So the actual number represented by the digit'strings is odd. Now this forces B to be odd, for if B were even, then 42050522(B) = 0 mod 2. So we have established that the number is odd and B is odd.Consider mod 3. To simplify things, let's reduce each digit mod 3, giving the numerals as 01201(A) and 12020222(B). Now look at the possibilities:A = 0 mod 3 ---> N = 1 mod 3 ... B = 0 mod 3 ---> N = 2 mod 3A = 1 mod 3 ---> N = 1 mod 3 ... B = 1 mod 3 ---> N = 2 mod 3A = 2 mod 3 ---> N = 2 mod 3 ... B = 2 mod 3 ---> N = 2 mod 3This proves that N = 2 mod 3 (no choice about that) and A = 2 mod 3 (because the mod 3 reduction of N must give the same answer independent of the base used in its expression).At this point, we know that A = 2 mod 3, B = 1 mod 2, and N = 5 mod 6.Consider mod 4. The numerals reduce to 33021(A) and 02010322(B), and the possibilities are:A = 0 mod 4 ---> N = 1 mod 4A = 1 mod 4 ---> N = 1 mod 4 ... B = 1 mod 4 ---> N = 2 mod 4A = 2 mod 4 ---> N = 1 mod 4A = 3 mod 4 ---> N = 3 mod 4 ... B = 3 mod 4 ---> N = 2 mod 4Here we run into trouble! There is no compatible choice, therefore, there CANNOT be any two bases A and B whatsoever, which will represent the same number N by these two particular strings of digits.Either I have made a calculation error (but I've rechecked, and I don't see one), or the original problem contains an error. Perhaps it is a typo in one of the digits? Perhaps instead it is a misinterpretation of what the number conversion operation actually did. Is it certain that it was simply a base conversion from one unspecified base to another?Hope this helps!--Stuart Anderson

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