Solve

Apply the First Isomorphism Theorem to prove that the quotient group $G/\ker(\varphi)$ is isomorphic to the image of $\varphi$.

Prove that every cyclic group of order $n$ is isomorphic to $\mathbb{Z}/n\mathbb{Z}$.

Use the Euclidean algorithm to find $\gcd(1071, 462)$ and express it as a linear combination.

Prove that the $3$-SAT problem is NP-complete using polynomial-time reduction from SAT.

Construct a deterministic finite automaton (DFA) that accepts all binary strings containing an even number of $1$s.

Use the Bolzano-Weierstrass Theorem to prove that every bounded sequence has a convergent subsequence.