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All divisible by 7 (Posted on 2008-11-04) Difficulty: 2 of 5
Imagine a rectangle divided into 3x4 squares, and put a digit in each square.
     | a | b | c | d |  A
     | e | f | g | h |  B
     | i | j | k | l |  C
       D   E   F   G
The number abcd is denoted by A, that is, A = 1000a + 100b + 10c + d, and the same for the other 2 horizontal numbers B and C.

The number aei is denoted by D, that is, D = 100a + 10e + i, and the same for the other 3 vertical numbers E, F and G.

Prove that if any 6 of these numbers (A, B, C, D, E, F, G) are divisible by 7, then the last number must also be divisible by 7.

See The Solution Submitted by pcbouhid    
Rating: 3.5000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re: Proof and Extensions | Comment 5 of 8 |
(In reply to Proof and Extensions by Steve Herman)

"I expect that this will work for any number of rows and columns, and any modulus, and any base. ".


Modulus 2: All three horizontal numbers (still base 10) are even. Only three of the vertical numbers are even. One is odd.

  Posted by Charlie on 2008-11-06 11:47:26
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