Question

the prime factorization of numbers can be used to find their gcf. follow the steps below to find the gcf of 24 and 40 using prime factorization.
1 write the prime factorization of 24 and 60.
2 circle the common prime factors.
3 find the product of the common prime factors.
find the gcf of each set of numbers using prime factorization.

1. 45 and 120
2. 30 and 84
3. 16, 20, and 48
4. candice is making flower arrangements. she has 56 white roses and 98 red roses. she wants to equally distribute the white roses and red roses to each arrangement. what is the greatest number of arrangements she can make? how many white roses and how many red roses are in each arrangement?

Explanation:

Step1: Prime - factorize 45 and 120

$45 = 3\times3\times5$, $120 = 2\times2\times2\times3\times5$

Step2: Identify common prime factors

Common prime factors are 3 and 5.

Step3: Calculate GCF

$GCF(45,120)=3\times5 = 15$

Step4: Prime - factorize 30 and 84

$30=2\times3\times5$, $84 = 2\times2\times3\times7$

Step5: Identify common prime factors

Common prime factors are 2 and 3.

Step6: Calculate GCF

$GCF(30,84)=2\times3 = 6$

Step7: Prime - factorize 16, 20 and 48

$16 = 2\times2\times2\times2$, $20=2\times2\times5$, $48=2\times2\times2\times2\times3$

Step8: Identify common prime factors

Common prime factors are $2\times2$.

Step9: Calculate GCF

$GCF(16,20,48)=4$

Step10: Prime - factorize 56 and 98

$56 = 2\times2\times2\times7$, $98=2\times7\times7$

Step11: Identify common prime factors

Common prime factors are $2\times7$.

Step12: Calculate GCF

$GCF(56,98)=14$
Number of arrangements is 14.
Number of white roses in each arrangement: $\frac{56}{14}=4$
Number of red roses in each arrangement: $\frac{98}{14}=7$

Answer:

1. GCF of 45 and 120 is 15
2. GCF of 30 and 84 is 6
3. GCF of 16, 20 and 48 is 4
4. Greatest number of arrangements is 14, 4 white roses and 7 red roses in each arrangement.