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Putnam 1995 A3

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July 1, 2014
Putnamcollege contests

Problem Statement

The number d1d2⋯d9d_1d_2\cdots d_9 has nine (not necessarily distinct) decimal digits. The number e1e2⋯e9e_1e_2\cdots e_9 is such that each of the nine 99-digit numbers formed by replacing just one of the digits did_i in d1d2⋯d9d_1d_2\cdots d_9 by the corresponding digit ei    (1≤i≤9)e_i \;\;(1 \le i \le 9) is divisible by 77. The number f1f2⋯f9f_1f_2\cdots f_9 is related to e1e2⋯e9e_1e_2\cdots e_9 is the same way: that is, each of the nine numbers formed by replacing one of the eie_i by the corresponding fif_i is divisible by 77. Show that, for each ii, di−fid_i-f_i is divisible by 77. [For example, if d1d2⋯d9=199501996d_1d_2\cdots d_9 = 199501996, then e6e_6 may be 22 or 99, since 199502996199502996 and 199509996199509996 are multiples of 77.]
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