I don't have a nice name for this!

Number Theory Level 3

( 5 2010 + 7 2010 ) m o d 74 = ? ({ 5 }^{ 2010 }+{ 7 }^{ 2010 })\bmod {74}=\ ?


The answer is 0.

Swapnil Das by Swapnil Das , 20, India

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3 solutions

Mathh Mathh
Aug 16, 2015

Use (for odd n n ):

a n + b n = ( a + b ) ( a n − 1 − a n − 2 b + a n − 3 b 2 − ⋯ + b n − 1 ) a^n+b^n=(a+b)\left(a^{n-1}-a^{n-2}b+a^{n-3}b^2-\cdots+b^{n-1}\right)

5 2010 + 7 2010 = 2 5 1005 + 4 9 1005 = 74 ( 2 5 1004 − 2 5 1003 49 + ⋯ + 4 9 1004 ) 5^{2010}+7^{2010}=25^{1005}+49^{1005}=74\left(25^{1004}-25^{1003}49+\cdots+49^{1004}\right)

6 Helpful 0 Interesting 0 Brilliant 0 Confused

Same method :)

Paola Ramírez - 5 years, 10 months ago
Dev Sharma
Aug 17, 2015

Using Euler and some calculation,

5^2010 = 27 mod74

7^2010 = 47 mod74

Add both, the answer is 0.

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Martin Raj Kumar
Aug 31, 2015

5^2010=25^1005 mod 74 = (-49)^1005 mod 74 =(-7^2)^1005 = -7^2010 So,-7^2010+7^2010 = 0 mod 74

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