If \(x\) and \(y\) are positive integers such that the greatest common factor of \(x^{2} y^{2}\) and \(x y^{3}\) is \(45,\) then which of the following could \(y\) equal? A. 45 B. 15 C. 9 D. 5 E. 3

Prime factorization is the process of breaking down a composite number into the product of its prime factors. Prime numbers are those that have exactly two distinct positive divisors: 1 and the number itself. For instance, the prime factorization of 45 is found by dividing the number by the smallest prime number that will go into it evenly, starting with 2 and then moving up the prime numbers. Since 45 is not even, we start with 3 and find that 45 is divisible by 3. Dividing 45 by 3 gives us 15, which is also divisible by 3, giving us 5, which is a prime number. So, the prime factorization of 45 is \( 3^2 \cdot 5 \).
Understanding prime factorization is crucial in finding the greatest common factor (GCF) between numbers and simplifies reducing fractions and solving problems involving divisors. It's a fundamental tool used in various mathematical applications, such as simplifying algebraic expressions and finding least common multiples.

Positive integers are all the whole numbers greater than zero. They do not include zero, fractions, decimals, or negative numbers. The positive integers are part of a group of numbers called the natural numbers, which play a fundamental role in all of arithmetic and number theory.

• They are used to count discrete quantities.
• They follow properties of being closed under addition and multiplication, meaning that adding or multiplying two positive integers always results in another positive integer.
• Understanding the properties of positive integers is essential for solving problems involving GCF, where we work exclusively with whole numbers to find the greatest integer that divides the given numbers without leaving a remainder.

In the context of the original exercise, the positive integers in question are the values of 'x' and 'y'. Determining the GCF helps identify the possible values of 'y' when multiplied by 'x' to achieve the desired GCF of 45.

The greatest common factor (GCF) of algebraic expressions is the largest expression that divides each of the given expressions without leaving a remainder. To find the GCF of two algebraic expressions, you need to:

• Factor each expression completely, the numeric part as well as the variable part.
• Identify common factors in both expressions.
• Choose the smallest power of each common factor.
• Multiply these together to get the GCF.

Example in Algebraic Terms

In the case of the original exercise, we were looking to find the GCF of \( x^2y^2 \) and \( xy^3 \). After examining the variable parts, it was observed that 'x' appears to at least the first power in both expressions and 'y' appears to at least the second power. As a result, the GCF was determined to be \( xy^2 \). This concept of evaluating the GCF is commonly used in factorization of algebraic expressions, simplifying fractions with variables, and solving equations or inequalities.