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Cube Difference Deduction (Posted on 2014-12-12) Difficulty: 3 of 5
Consider the seven positive integers: 123, 1234, 12345, 123456, 1234567, 12345678 and 123456789.

How many of these seven numbers are expressible as the difference of cubes of two positive integers?

No Solution Yet Submitted by K Sengupta    
Rating: 5.0000 (1 votes)

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Solution Rule of cubes Comment 2 of 2 |
All cubes are 0, 1, or 8 mod 9. The difference of two cubes has to be 0, 1, 2, 7, or 8 mod 9.

123=6 mod 9
1234=1 mod 9
12345=6 mod 9
123456=3 mod 9
1234567=1 mod 9
12345678=0 mod 9
123456789=0 mod 9

Now, we know that it has to be 1234, 1234567, 12345678, or 123456789. The answer is 1234567=642^3-641^3.


  Posted by Math Man on 2014-12-20 20:39:08
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