Compute large powers modulo n
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.23 Determine the last two digits of $3^{3^{100}}$. (Find $3^{100} \mod {\varphi(100)}$ and use Exercise 3.2.22.) Solution:…
Use Lagrange’s Theorem to prove Euler’s Theorem
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.22 Use Lagrange’s Theorem in the multiplicative group $G = (\mathbb{Z}/(n))^\times$ to prove Euler’s Theorem: if $\mathsf{gcd}(a,n)…