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Divisibility by 7 (Posted on 2005-05-22) Difficulty: 2 of 5
(2222^5555 + 5555^2222) is or isn't divisible by 7 ?

Just pencil and paper.

See The Solution Submitted by pcbouhid    
Rating: 2.0000 (3 votes)

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Solution divisible by 7? | Comment 3 of 8 |

Is 22225555 + 55552222 divisible by 7?

22225555 + 55552222 = 35555 + 42222 (mod 7). There are only 7 residue classes modulo 7. Therefore, the series of powers of a given number (modulo 7) is bound to be cyclic (of period 6). For example, 32 = 2 (mod 7), 33 = 6 (mod 7), 34 = 4 (mod 7), 35 = 5 (mod 7), 36 = 1 (mod 7), 37 = 3 (mod 7), and so on: 3,2,6,4,5,1,3,... Note that 5555 = 5 (mod 6). Therefore, 22225555 = 35 (mod 7) = 5(mod 7).

Similarly, 55552222 = 42222 (mod 7). Powers of 4 modulo 7 form a cycle of period 3: 4,2,1,4,... Hence, 42222 (mod 7) = 42222 ( mod 3)(mod 7) = 42 (mod 7) = 2 (mod 7).

Adding up 5 + 2 gives a number divisible by 7. So is 22225555 + 55552222.


  Posted by heather on 2005-05-23 14:02:42
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