Problem #PRU-100602

Problem

Two numbers are given in terms of their prime factorizations: $a=2^3\times 3^2\times 5\times {11}^2\times {17}^2$ and $b=2\times 5^3\times 7^2\times 11\times 13$.

a) What is the greatest common divisor $\gcd (a,b)$ of these numbers?

b) What is their least common multiple $\mathrm{lcm}(a,b)$?

c) Write down the prime factorization of $\gcd (a,b)\times \mathrm{lcm}(a,b)$. Then write the prime factorization of $a\times b$. What do you notice?