Question

Find the remainder when 2^5^9 + 5^9^2 + 9^25 is divided by 11.

Answer 1

5

Answer 2

how did you do it?? I found 8...

Answer 3

You need to to each expression individually. So you need to find the remainder when 2^5^9 is divided be 11, when 5^9^2 is divided by 11, and when 9^25 is divided by 11. Let's do 2^5^9 first.

We need to solve \[\large 2^{5^9} \mod 11\]To do this, we need to find \(5^9 \mod 10\). Fortunately, any power of 5 is 5 mod 10. So now we need to calculate \[\large2^5 \mod 11\]Since \(2^5=32\) this is equal to 10 mod 11.

Answer 4

Now let's move on to \[\large 5^{9^2} \mod 11\]Here we need to find \(9^2 \mod 10\). We calculate this to be \(81\equiv 1 \mod 10\). So we have \[\large 5^{9^2} \equiv 5^1 \equiv 5\mod 11\]

Answer 5

Finally, we need \[\large 9^{25} \mod 11\]Like above, we find \(25\mod10\) to be 5. So we calculate \[\large 9^{5} \equiv 59049 \equiv 1 \mod 11\]

Answer 6

We finally add up our values, and take the remainder when divided by 11. We get \[10+5+1\equiv16\equiv 5 \mod 11\]So our answer is 5.

Answer 7

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Answer 8

You're welcome.