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anonymous

  • 4 years ago

Fool's problem of the day, Find the remainder when \( 44^8 \) is divided by \( 119 \). PS:This problem is originated from one of the myininaya's reply at OS feedback chat.

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  1. RagingSquirrel
    • 4 years ago
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    eh... i hate these

  2. barrycarter
    • 4 years ago
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    67 by brute force: http://fwd4.me/0lMq

  3. anonymous
    • 4 years ago
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    hehe

  4. anonymous
    • 4 years ago
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    For myin, the constraint is higher \( 44^{86} \) ;-)

  5. anonymous
    • 4 years ago
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    Its extremely easy. Here: 44^2 = 1936 = 32 mod 119 implies 44^4 = 32^2 = 1024 = 72 mod 119 implies 44^8 = 72^2 = 5184 = 67 mod 119

  6. anonymous
    • 4 years ago
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    Sure Aron, now try the one with higher constraints.

  7. KingGeorge
    • 4 years ago
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    Well, the second problem can be reduced to solving for x in\[44^{10} x \equiv 1 \mod \; 119\]using Euler's totient function. From there, we can find that\[x=60\]Unfortunately, I still need the help of Wolfram for that last step :(

  8. anonymous
    • 4 years ago
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    Okay,here it is : 86 = 2*43., 44 = 4*11 44^8 = 67 mod 119 implies 44^40 = 67^ 5 = 16 mod 119 implies 44^43 = 64*16= 72 mod 119 implies 44^86 = 72^2 = 5184 = 67 mod 119.Done!

  9. anonymous
    • 4 years ago
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    Aron, you are right apart from the fact the the remainder is not 67.

  10. KingGeorge
    • 4 years ago
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    Alternatively, we know that \[44^{86}=44^{64}*44^{16}*44^4*44^2\]and by computing successive squares of 44 modulo 119 we can also find that \[44^{86} \equiv 60 \mod \: 119\]

  11. KingGeorge
    • 4 years ago
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    This is the the same method Aron used for finding \[44^8 \mod \: 119\]

  12. anonymous
    • 4 years ago
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    That's right KIngGeorge, however it's extremely tedious when you don't have any electronic help.

  13. KingGeorge
    • 4 years ago
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    Is there a way to do it quickly without electronic help?

  14. anonymous
    • 4 years ago
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    There always is :)

  15. anonymous
    • 4 years ago
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    You can use Euler's & Fermat's Theorem!

  16. KingGeorge
    • 4 years ago
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    How so?

  17. anonymous
    • 4 years ago
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    @Fool I am sorry! I just checked the remainder to be 60.

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