Friday’s problem was to solve the following cryptarithm or alphametic, where each letter represents a digit 0-9.
EVE / DID = .TALKTALK . . .
Neither EVE nor DID has a leading zero, and EVE / DID has been reduced to its lowest terms. The result of the division is a recurring decimal with a repeating pattern of four digits.
There is one unique solution.
To resolve a recurring decimal you put the recurring part over the same number of nines. In this case, the four letters of TALK go over 9999, which gives
EVE / DID = TALK / 9999
From this DID must be a factor of 9999. The prime factors of 9999 are 3, 3, 11 and 101, so the three possible options for DID are 101, 303 and 909.
If DID is 101 then
EVE / 101 = TALK / 9999 and
99 * EVE = TALK
The smallest number that EVE can be is 101, as anything larger gives a five-digit product with 99. EVE can’t be 101 as DID is 101, so EVE can’t be 101.
If DID is 909 then
EVE / 909 = TALK / 9999 and
11 * EVE = TALK
EVE multiplied by 11 must give a number beginning with E, and TALK does not, so DID can’t be 909.
Thus DID must be 303, and because EVE must be smaller than DID, it must start with 1 or 2. The fourteen possibilities for EVE are 121, 141, 151, 161, 171, 181, 191, 212, 242, 252, 262, 272, 282 and 292.
Working through these, the only one that produces a four-digit recurring decimal, with no letter (number) clashes, is 242.
242 / 303 = .79867986 . . .