Can someone please explain this problem.
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Module 5 Day 3 Your Turn Part 3 Mini-Question Solution
It says that this number is odd, when it is even because 8^7^6^5 and so on can be expressed as 8^(7!), and we know that even*even=even.
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Very close! It would only be equal to 8^(7!) if there were parentheses around each term like this:
If there are no parentheses, the powers cannot be multiplied, and so it equals this:
Since the exponent on the 8 is a power of 7, which is odd, the answer is equal to 8^{1} mod 9. -
@v4913 said in Can someone please explain this problem.:
Very close! It would only be equal to 8^(7!) if there were parentheses around each term like this:
If there are no parentheses, the powers cannot be multiplied, and so it equals this:
Since the exponent on the 8 is a power of 7, which is odd, the answer is equal to 8^{1} mod 9.o pro v4913 post o
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@v4913 Ok, thanks! I also have another question about this problem. When it is odd, why is the answer -1 mod 9?
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@neatlobster @v4913 Thank you
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@neatlobster Good question! Since 8 = -1 mod 9, and the powers of -1 go -1, 1, -1, 1, etc, all (-1)^{even} = 1 mod 9 and all (-1)^{odd} = -1 mod 9. Since we are talking about modulos, and 8 is the same as -1 mod 9, this means (8)^{even} = 1 mod 9, and (8)^{odd} = -1 mod 9.
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@v4913 Since there isn't any parentheses, what does this formula mean?
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@amusingminnow Good question! It's highly non-obvious, but the rules of operations say that when there are stacked powers with no parentheses, we work from the top down, so for example,
We can continue this pattern to evaluate the entire expression.(Sorry for the bad formatting lol)