How to apply the Chinese remainder theorem here?

Question

Mathematics

Calculate last two digits in decimal notation 7^9^9

Answer 1

Um, for starters, (7^9)^9 or 7^(9^9)?
I don’t understand about the Chinese theorem, but the powers of the seven have remainders from dividing by 100 (that is, the last two digits): 01, 07, 49, 43 and so on in a circle. It is necessary to determine the remainder of the division by 4 exponents - in any case, it is equal to 1, like any degree of 9. So, the answer is -07

Answer 2

Find the remainder after dividing the item you are looking for by 4 and by 25. Next, apply the Chinese remainder theorem.
It's easier, because the remainder of dividing a^(b^c) by a power of a prime number can be found by Euler 's little theorem :
a^(b^c) = a ^ (b^c % (p-1)p^(n -1)) mod p^k.
Those. 7^9^9 = 7^((9^9) % 20) mod 25
9^9 % 20 can already be calculated via logarithmic exponentiation.
Similarly for 4.
True, here it is possible without the remainder theorem. We just need to immediately apply Euler's theorem. phi(100) only needs to be carefully calculated. And the numbers will be a little more, but still lifting.