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| ten - nine = 1 (Posted on 2009-04-23) |
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Solve this alphametic, where ten and nine are respectively numbers in base 10 and base 9 notation. Each of the small letters in bold represents a different digit, and none of the numbers contains any leading zero.
(ten)10 - (nine)9 = 1
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Submitted by K Sengupta
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Rating: 5.0000 (1 votes)
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Solution:
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If n ≥2, then :
(ten)10 = (nine)9 + 1 > (1000)10, which is a contradiction.
Thus, n=1, so that:
(te1)10 = (1i1e)9 + 1
-> 100*t + 9*e – 81*i - 738 = 0
-> t = 0 (mod 9)
Since we cannot have any leading zero, it follows that t = 9
Hence, 81*i – 9*e = 162
Or, 9i – e = 18, which gives: (i, e) = (2,0)
Therefore, ten = 901 and, nine = 1210 is the only possible solution to the given problem.
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*** Adapted from JRM 1034 by H. Everett Moore.
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