Divisibility by 37
Date: 11/08/97 at 04:22:37 From: P.A. van Renselaar Subject: Rotation sum Take a number of 3 figures, and add to that its "rotation": 257 + 572 + 725 = 1554 The "rotation sum" can be divided by 37. (1554 = 42 x 37) Take a look at this one: 899 + 998 + 989 = 78 x 37 360 + 603 + 036 = 27 x 37 Prove that this can be said for every 3-digit number.
Date: 11/08/97 at 09:21:53 From: Doctor Anthony Subject: Re: Rotation sum The number 37 can be expressed as 111/3, so 16 x 37 = 16 x 111/3 = (5 + 1/3) x 111 = 592 = 555 + 37 In fact all 3-digit multiples of 37 will be of the two possible forms: aaa + 37 or aaa - 37 Now if you take a number like 592 = 500 + 90 + 2 and rotate the order of the digits, you get 259 = 200 + 50 + 9 Now to get this from the original you +300 + 40 - 7 = 333, so we have: 259 = 592 - 333 = M(37) - M(37) = M(37) where M(37) = multiple of 37. You can see that any rotation of the digits of a multiple of 37 is done by either adding or subtracting 333, and since 333 is a multiple of 37, the result remains a multiple of 37. Check with some other multiple of 37, say 14 x 37 = 518 (= 555-37) 518 - 333 = 185 and 518 + 333 = 851 both being rotations of the digits. Another example 17 x 37 = 629 (= 666-37) 629 - 333 = 296 and 629 + 333 = 962 and once again these are cyclic rotations of the original digits. The question is why this happens. We showed that every 3-digit multiple of 37 is of the form: aaa + or - 37 Taking the +37 case we therefore have the digits in the following pattern a a+3 a+7 now add 333 a+3 a+6 a+10 but this means carry a 1 to the tens column, giving a+3 a+7 a So the digits are now a+3 a+7 a instead of a a+3 a+7 As you can see, there has been a rotation of the digits. What happens if we have aaa - 37 ? digits are a a-3 a-7 and if there is a carry a a-4 a+3 now subtract 333 a-3 a-7 a and applying the carry a-4 a+3 a So instead of digits a a-4 a+3 we get a-4 a+3 a and again it is a cyclic rotation of digits but now in the reverse direction. The fact that multiples of 37 are aaa + 37 or aaa-37 is the reason for this odd pattern. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/
Date: 07/25/2001 at 22:58:56 From: P.A. van Renselaar Subject: Divisibility by 37; another solution I don't think your last explanation was easy for many people. Therefore I offer this solution: Take the number 254. Look at the rotations of its digits and note that 254 + 542 + 425 is divisible by 37 (1221 / 37 = 33). Now look at the general form of the number abc. (The other rotations are bca and cba.) It is easy to see that the number abc can be written: 100a + 10b + c and so the other rotations are 100b + 10c + a 100c + 10a + b Adding these up gives: 100(a+b+c) + 10(a+b+c) + (a+b+c) This form can be written as 111(a+b+c) Note that 111 is divisible by 37. QED.
Date: 07/30/2001 at 10:22:50 From: Doctor Peterson Subject: Re: Divisibility by 37; another solution My impression, looking at Dr. Anthony's original answer, is that he was really answering a somewhat different (and harder) question: prove that every 3-digit multiple of 37, when rotated, remains a multiple of 37. I think that's why it's hard to follow. Your approach to proving that the sum of all rotations of any three-digit number is divisible by 37, is just what I would probably have done, and is a very neat, straightforward proof. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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