Problem 5. [15 points] Let p, q, and r be distinct primes. Prove that there exist integers a, b, and c such that: a·(p)+b·(qr)+c·(rp)=1 Hint: First, consider linear combinations of just pq and qr. Solution. Since gcd(pq, qr)=g, there exist integers s and t such that s(pq)+t(ar)=q Now gcd(q, rp)=l, so there exist integers u and v such that uq+v(rp) Therefore u(s(py)+t(qr)+v(rp)=(us)(pq)+()(qr)+v(rp)=1Quiz 1 8 Problem 5. [15 points] Let p, q, and r be distinct primes. Prove that there exist integers a, b, and c such that: a · (pq) + b · (qr) + c · (rp) = 1 (Hint: First, consider linear combinations of just pq and qr.) Solution. Since gcd(pq, qr) = q, there exist integers s and t such that: s(pq) + t(qr) = q Now gcd(q, rp) = 1, so there exist integers u and v such that: uq + v(rp) = 1 Therefore: u(s(pq) + t(qr)) + v(rp) = (us)(pq) + (ut)(qr) + v(rp) = 1