Question

What is the remainder when 3^12 + 5^12 is divided by 13?

Answer 1

3^12 + 5^12
= 27^4 + 25^6

now 27 = 1 mod 13 =>27^4 = 1 mod 13
and 25 = -1 mod13 =>25^6 = 1 mod 13

bod leave remainder 1 each,,hence net remainder will be 1+1 or 2..

Answer 2

or one can use fermat little theorem :)

Answer 3

Fermat's little theorem states that if \(p\) is a prime number, then for any integer \(a\)\[a^{p-1} \equiv1 \ \ \ \text{mod} \ p\]

Answer 4

http://en.wikipedia.org/wiki/Fermat's_little_theorem

Answer 5

would you just show me that in this question?

Answer 6

sure p=13 is a prime number so \[3^{13-1}=3^{12} \equiv1 \ \ \ \text{mod} \ 13\]\[5^{13-1}=5^{12} \equiv1 \ \ \ \text{mod} \ 13\]so\[3^{12}+5^{12} \equiv2 \ \ \ \text{mod} \ 13\]

Answer 7

learnt something new @mukushla ..thanks..

Answer 8

np man :)