Maximum Number of Sets of Books and Notebooks for Distribution

The problem asks us to find the maximum number of identical sets of books and notebooks that can be made from 2 dozen books and 16 notebooks, such that no items are left over after distribution. This type of problem requires us to find the Greatest Common Divisor (GCD) of the number of books and the number of notebooks.

Converting Dozens to Units

First, let's determine the total number of books. One dozen is equal to 12 items. We have 2 dozen books.

• Number of books = 2 dozen = $2 \times 12 = 24$ books.
• Number of notebooks = 16 notebooks.

Finding the Maximum Number of Identical Sets

To make the maximum number of identical sets with 24 books and 16 notebooks, without any remaining items, we need to find the largest number that can divide both 24 and 16 exactly. This number is the Greatest Common Divisor (GCD) of 24 and 16.

Calculating the GCD of 24 and 16

We can find the GCD using the prime factorization method or by listing factors.

Using Prime Factorization:

• Find the prime factorization of 24: $24 = 2 \times 12 = 2 \times 2 \times 6 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3^1$
• Find the prime factorization of 16: $16 = 2 \times 8 = 2 \times 2 \times 4 = 2 \times 2 \times 2 \times 2 = 2^4$

To find the GCD, we take the common prime factors raised to the lowest power they appear in either factorization. The only common prime factor is 2. The lowest power of 2 is $2^3$.

GCD(24, 16) = $2^3 = 8$.

Using Listing Factors:

List all the factors of 24 and 16.

Identify the common factors: 1, 2, 4, 8.

The greatest among the common factors is 8.

So, GCD(24, 16) = 8.

Result: Maximum Number of Sets

The Greatest Common Divisor of 24 and 16 is 8. This means that the maximum number of identical sets of books and notebooks that can be made is 8.

Each of these 8 sets will contain:

• Number of books per set = $\frac{24 \text{ books}}{8 \text{ sets}} = 3$ books per set
• Number of notebooks per set = $\frac{16 \text{ notebooks}}{8 \text{ sets}} = 2$ notebooks per set

Thus, a maximum of 8 sets can be made, with each set having 3 books and 2 notebooks.

Revision Table: Books and Notebooks Distribution Maximum Sets

Additional Information: Understanding GCD in Real-World Problems

The Greatest Common Divisor (GCD), sometimes called the Highest Common Factor (HCF), has practical applications in various scenarios, particularly in problems involving division into equal parts or grouping items.

• Grouping: Finding the largest number of identical groups you can make from different quantities of items.
• Measurement: Finding the largest possible length of a measuring tape that can exactly measure two or more different lengths.
• Arrangement: Arranging items in rows or columns where each row/column has the same number of items of different types.

In this problem, applying the concept of GCD allowed us to determine the maximum number of identical packages (sets) we could create using all the available books and notebooks.