Question

1. reasoning you have 50 blueberry scones and 75 cranberry scones. you want to make as many identical bags as possible. each bag should have an equal number of blueberry scones and an equal number of cranberry scones. what is the greatest number of bags you can fill? explain.
2. make sense and persevere the prime factorizations of a and b are shown. find the value of n that needs to be listed as a prime factor of b so that the greatest common factor (gcf) of a and b is 9.

prime factorization of a: 3 × 3 × 3
prime factorization of b: 2 × 2 × 3 × n

Explanation:

Problem 31

Step1: Find prime factors

Find prime factors of \(50 = 2\times5\times5\) and \(75=3\times5\times5\).

Step2: Determine GCF

The greatest common factor (GCF) is the product of common prime factors. Common prime factor is \(5\times5 = 25\).

Step1: Analyze GCF formula

The GCF of two numbers is the product of the common prime factors with the lowest exponent. Given \(A = 3\times3\times3\) and \(B=2\times2\times3\times n\), and \(GCF(A,B)=9 = 3\times3\).

Step2: Solve for \(n\)

Since the GCF has two \(3\)s, and \(B\) already has one \(3\), then \(n = 3\) (so that \(B\) has two \(3\)s as part of common factors with \(A\)).

Answer:

The greatest number of bags you can fill is \(25\). Because the GCF of \(50\) and \(75\) is \(25\), which means we can divide both \(50\) blueberry scones (\(50\div25 = 2\) per bag) and \(75\) cranberry scones (\(75\div25=3\) per bag) evenly into \(25\) bags.

Problem 32