List the first \(10\) prime numbers and write the prime decomposition of \(2910\).
The number \(a\) has a prime factorization \(2^3 \times 3^2 \times 7^2 \times 11\). Is it divisible by \(54\)? Is it divisible by \(154\)?
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?
The prime factorization of the number \(b\) is \(2 \times 5^2 \times 7 \times 13^2 \times 17\). The prime factorization of the number \(c\) is \(2^2 \times 5 \times 7^2 \times 13\). Is the first number divisible by the second one? Is the product of these two numbers, \(b \times c\), divisible by \(49000\)?