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Old 2007-05-13, 21:14   #1
davar55
 
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Default Divisible by 7

Show that 22225555 + 55552222 is divisible by 7.

More generally, when is ab + ba prime?
(This part I don't have a completely general answer to.)
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Old 2007-05-13, 22:05   #2
akruppa
 
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2222 - 2100 = 122, 122 - 70 = 52, 52 - 49 = 3, so 2222%7 = 3. 3^6 == 1 (mod 7), 5555 - 4800 = 755, 755 - 720 = 35, 35 - 30 = 5, so 5555%6 = 5. 2222^5555 = 3^5 == 5 (mod 7). (The mod reduction is a bit long-winded, but I wanted to do this without electronic aid)

Same for 5555^2222 == 4^2 == 2 (mod 7).

5 + 2 == 0 (mod 7)


Iirc, Paul Leyland has something to say about the more general question.

Alex

Last fiddled with by akruppa on 2007-06-04 at 15:51 Reason: fixed /spoiler
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Old 2007-05-14, 07:44   #3
xilman
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Quote:
Originally Posted by akruppa View Post
Iirc, Paul Leyland has something to say about the more general question.

Alex
Indeed. http://www.leyland.vispa.com/numth/primes/xyyx.htm is where it's at.

Note that site badly needs updating with more results sent to me relatively recently.


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Old 2007-05-14, 22:05   #4
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Yeah, it's pretty clear that you use the properties of modulas and exponentiation together.

Last fiddled with by nibble4bits on 2007-05-14 at 22:05
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