<s>[INST] Determining the last two digits of $229^{10} +37^{10}$ Determine the last two digits of: $229^{10}+37^{10}.$
I do not want to use the Euler-totient function or the carmichael function please!
Thanks
 [/INST] $37^{10}\equiv 7^{10}\equiv 49\pmod{100}$
$229^{10}\equiv 9^{10}\pmod {100}$
$\begin{align}9^3\equiv 29\pmod {100}, 9^2\equiv-19\pmod{100} & \implies 9^5\equiv -(29\cdot19)\pmod{100}\\&\implies 9^5\equiv -551\pmod{100}\\&\implies 9^5\equiv -51\pmod{100}\\&\implies 9^{10}\equiv 1\pmod{100}\end{align}$
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