Question

Calculate the total number of positive, even common divisors shared by x=2⁵⋅3²⋅5¹⁵⋅7³⋅11¹² and y=2⁴⋅3⋅5¹⁰⋅7⁵⋅11⁵

Answer 1

To calculate the total number of positive, even common divisors shared by x and y, find the prime factorization of both numbers and consider the even exponents of the common prime factors.

Step-by-step explanation:

In order to calculate the total number of positive, even common divisors shared by the two numbers, we need to find the prime factorization of both x and y.

The prime factorization of x is
`2^5 * 3^2 * 5^15 * 7^3 * 11^12.`

The prime factorization of y is
`2^4 * 3 * 5^10 * 7^5 * 11^5.`

To find the common divisors, we need to consider the exponents of the prime factors that are present in both x and y. Since we are looking for even divisors, we need to consider the even exponents.

The common divisors with even exponents are
`2^4, 2^2, 2^0` there are three positive, even common divisors between x and y.