document.write( "Question 1207672: For an integer $n,$ let $f(n)$ be the remainder when $n^8 + n^{16}$ is divided by $5.$ Compute $f(0) + f(1) + f(2) + f(3) + f(4).$ \n" ); document.write( "

Algebra.Com's Answer #845675 by greenestamps(13200) $\"\"$ $\"About$

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\n" ); document.write( "For n=0,1,2,3,4 determine the pattern of the powers of n mod 5:

\n" ); document.write( "n=0: n to any power is 0; the remainder when divided by 5 is always 0
\n" ); document.write( "n=1: n to any power is 1; the remainder when divided by 5 is always 1
\n" ); document.write( "n=2: the pattern is 2, 4 (=-1), -2, 1 (cycle length 4)
\n" ); document.write( "n=3: the pattern is 3, 9 (=-1), -3, 1 (cycle length 4)
\n" ); document.write( "n=4: the pattern is 4 (=-1), 1 (cycle length 2)

\n" ); document.write( "f(n) is equal to (n^8+n^16) mod 5.

\n" ); document.write( "The lengths of the cycles of the remainders are 1, or 2, or 4; 8 and 16 are both multiples of all of those. So both n^8 mod 5 and n^16 mod 5 will be the last number in the pattern for each value of n.

\n" ); document.write( "The last numbers in the patterns are 0 for n=0 and 1 for all other values of n. So

\n" ); document.write( "f(0) = 0+0 = 0
\n" ); document.write( "f(1) = 1+1 = 2
\n" ); document.write( "f(2) = 1+1 = 2
\n" ); document.write( "f(3) = 1+1 = 2
\n" ); document.write( "f(4) = 1+1 = 2

\n" ); document.write( "f(0)+f(1)+f(2)+f(3)+f(4) = 0+2+2+2+2 = 8

\n" ); document.write( "ANSWER: 8

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