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 2007-05-13, 21:14 #1 davar55     May 2004 New York City 3×17×83 Posts 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.)
 2007-05-13, 22:05 #2 akruppa     "Nancy" Aug 2002 Alexandria 9A316 Posts 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
2007-05-14, 07:44   #3
xilman
Bamboozled!

"πΊππ·π·π­"
May 2003
Down not across

101001101110102 Posts

Quote:
 Originally Posted by akruppa 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.

Paul

 2007-05-14, 22:05 #4 nibble4bits     Nov 2005 18210 Posts 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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