## 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:…

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:…

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)…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.15 Let $p$ be a prime and let $n$ be a positive integer. Find the order of…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.16 Use Lagrange’s Theorem in the multiplicative group $(\mathbb{Z}/(p))^\times$ to prove Fermat’s Little Theorem: if $p$ is…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.5 Find the number of generators for $\mathbb{Z}/(49000)$. Solution: The number of generators of $\mathbb{Z}/(n)$ is $\varphi(n)$,…