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 \(\mathrm{gcd}(a,b)\) of these numbers?
b) What is their least common multiple \(\mathrm{lcm}(a,b)\)?
c) Write down the prime factorization of \(\mathrm{gcd}(a,b) \times \mathrm{lcm}(a,b)\). Then write the prime factorization of \(a \times b\). What do you notice?