what is the remainder when
17^2^111110253 is divided by 21
please show working
#thanks
To find the remainder when \(17^{2^{111110253}}\) is divided by 21, we need to simplify the exponent first.
First, let's calculate the remainder for powers of 17 when divided by 21, and look for any patterns:
\(17^1\) = 17 (remainder = 17)
\(17^2\) = 289 (remainder = 1)
\(17^3\) = 4913 (remainder = 4)
\(17^4\) = 83521 (remainder = 16)
\(17^5\) = 1419857 (remainder = 13)
\(17^6\) = 24137569 (remainder = 1)
\(17^7\) = 410338673 (remainder = 4)
Notice that a pattern emerges. Starting with the power 2, the remainders repeat every 6 powers: 1, 4, 16, 13, 1, 4, ...
The remainder of \(17^{2^{111110253}}\) divided by 21 will be the same as the remainder of \(17^{2^{111110253 \mod 6}}\) divided by 21.
Now let's simplify the exponent, \(2^{111110253 \mod 6}\):
\(111110253 \mod 6 = 5\)
So, we need to calculate the remainder when 2 is raised to the power of 5:
\(2^5 = 32\) (remainder = 11)
Now we have the remainder of the exponent: \(111110253 \mod 6 = 5\) and the remainder of the base: \(2^5 = 32\) (remainder = 11).
Finally, we need to calculate the remainder when 17 raised to the power 11 is divided by 21. This requires one more step:
\(17^{11} \mod 21\):
\(17^1\) = 17 (remainder = 17)
\(17^2\) = 289 (remainder = 1)
\(17^3\) = 4913 (remainder = 4)
\(17^4\) = 83521 (remainder = 16)
\(17^5\) = 1419857 (remainder = 13)
\(17^6\) = 24137569 (remainder = 1)
\(17^7\) = 410338673 (remainder = 4)
\(17^8\) = 6975757449 (remainder = 16)
\(17^9\) = 118587876497 (remainder = 13)
\(17^{10}\) = 2015993900449 (remainder = 1)
\(17^{11}\) = 34271896307633 (remainder = 4)
Therefore, the remainder of \(17^{2^{111110253}}\) divided by 21 is 4.