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**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**.

- Resolve each of the given
**numbers**into its**prime factors** - Highlight every
**common factor** **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**:

- Resolve each of the given
**numbers**into its**factors** - Identify the
**highest common factor**among them all - 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:

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

\[GCF = 12\]