EVE/DID = .TALKTALKTALK...Date: 11/21/2000 at 16:23:50 From: Stuart Subject: Recurring decimals The problem is eve/did = .talktalktalk... Each letter corresponds to a different number, and the fraction eve/did is in lowest form. So far, this is what I have done: talk/9999 = .talktalktalk... = eve/did Dividing 9999 by 3 and 11 gives 303. By trial and error I have found that 242/303 gives a four-digit repeating decimal with no digits repeated in the fraction or decimal. However, I think this was more through luck than judgment. Also, I have to find if there are more solutions and, if not, why. Date: 11/21/2000 at 17:02:13 From: Doctor Rob Subject: Re: Recurring decimals Thanks for writing to Ask Dr. Math, Stuart. Rewrite this in the form 9999*EVE = DID*TALK 3*3*11*101*EVE = DID*TALK Now 101 must be a divisor of the right side. If 101 is a divisor of TALK, then you would have T = L and A = K, so that is impossible. Thus 101 is a divisor of DID. The quotient is D, and this implies that D*101 = D0D = DID, so I = 0. This reduces the condition to 3*3*11*EVE = D*TALK Now E >= 1, so the left side is at least 121*99 = 11979. This implies that D >= 2. But D cannot have a factor in common with EVE, or else EVE/DID would not be in lowest terms. Thus D must be a divisor of 99, so D = 3 or 9. If D = 9, then 11*EVE = TALK would imply that E = K, which is impossible. Thus D = 3, and DID = 303. This reduces the equation to 33*EVE = TALK This implies that E <= 2. If E = 1, that would make K = 3 = D, an impossibility. Thus E = 2. That reduces the equation to 6666 + 330*V = TALK so K = 6. Then 666 + 33*V = TAL The numbers 0, 2, 3, and 6 are already taken. Thus there are the following possibilities for V: 1, 4, 5, 7, 8, or 9. Try each to see if you get any duplications of letters. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ |
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