Simon is making wreaths to sell. He has 60 bows, 36 silk roses, and 48 silk carnations.

All wreaths have the same items, and he needs to put the same number of items in each. How many items will come in each wreath?

The aim of the question is to find the GCF for the given numerical numbers.

The basic concept behind this problem is the knowledge of the Greatest Common Factor.

GCF stands for Greatest Common Factor, defined as the greatest Factor common between the required numbers for which GCF is to be determined. It is the greatest positive number that is divisible by all given numbers. GCF can be determined between 2 or more than 2 numbers.

Here is the Step-by-Step Procedure to calculate the $GCF$ $Greatest$ $Common$ $Factor$ of two or more numbers by using the method of Prime Factorization.

1. Resolve each of the given numbers into its prime factors
2. Highlight every common factor
3. Multiply all the common factors to get $GCF$

For smaller numbers, the method of multiplies is more convenient. Following is the Step-by-Step Procedure to find the $GCF$ $Greatest$ $Common$ $Factor$ by using the method of multiplies:

1. Resolve each of the given numbers into its factors
2. Identify the highest common factor among them all
3. The highest common factor is our required GCF

The $GCF$ of two or more polynomial expressions is represented by the expression or factor having the greatest power such that all the given polynomials can be divisible by that factor. It is explained as follows:

$(i)$ Resolve each of the given polynomial expressions into its factors.

$(ii)$ The factors having the highest power, or the highest degree in each expression will be multiplied to calculate the $GCF$ for the given polynomial expression.

$(iii)$ In the presence of numerical coefficients or constants, calculate their $GCF$ also.

$(iv)$ Multiply the $GCF$ of factors with the highest power and $GCF$ of coefficients or constants to calculate the $GCF$ of given polynomial expressions.

Here, we will find the $GCF$ by using the method of multiples i.e. finding the common multiples between the given numbers and then selecting the greatest among them as the $GCF$ for that pair.

Expert Answer

Given in the question, we have:

$Bows\ = 60$

$Silk\ roses\ = 36$

$Silk\ carnations\ = 48$

Now the factors of the given numbers, we write them as:

\[60=1,2,3,4,5,6, 10, 12, 15,20,30,60\]

\[36=1,2,3,4,6,9,12,18,36\]

\[48=1,2,3,4,6,8,12,16,24,48\]

As we can see, $12$ is the highest common factor in all, so $GCF=12$

\[GCF =12\]

Numerical Results:

So the required number of items are:

$Bows\ = 5$

$Silk\ roses\ = 3$

$Silk\ carnations\ = 4$

For a total of $12$ items in each wreath.

Example:

Find out the $GCF$ for the following numbers by using Prime factorization method.

\[60, 36, 48\]

Solution:

The prime factors of $60$, $36$ and $48$ will be:

\[60\ = 2 \times 2 \times 3 \times 5\]

\[36\ = 2 \times 2 \times 3 \times 3\]

\[48\ = 2 \times 2 \times 2 \times 2 \times 3\]

So, the common factors will be:

\[GCF = 2 \times 2 \times 3\]

\[GCF =  12\]

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