school is repairing is package using 180 chocolates 240 pencils and 300 research contains the same number of each item with no items leftover What is the greatest number of identical packets that can be made how many chocolates release packet contain how many pencils and races village packet contain

Answer: The greatest number of identical packages that can be made is 60. Each package will contain 3 chocolates, 4 pencils, and 5 erasers.

Step-by-step solution

Step 1: Identify the required mathematical operation

To divide the $180$ chocolates, $240$ pencils, and $300$ erasers into identical packages with none left over, the number of packages must be a common factor of all $3$ quantities. To make the greatest number of identical packages, we need to find the Highest Common Factor ( $\operatorname{HCF}$ ) of $180$ , $240$ , and $300$ .

Step 2: Find the prime factorisation of each number

Using the Fundamental Theorem of Arithmetic, we express each number as a product of its prime factors. For $180$ , it factors into $2^2 \times 3^2 \times 5$ . For $240$ , it is $2^4 \times 3 \times 5$ . For $300$ , it is $2^2 \times 3 \times 5^2$ .

Step 3: Calculate the HCF

The $\operatorname{HCF}$ is the product of the smallest power of each common prime factor involved in the numbers. Taking the minimum exponents for each prime: $2^2 = 4$ , $3^1 = 3$ , and $5^1 = 5$ , giving an $\operatorname{HCF}$ of $4 \times 3 \times 5 = 60$ . Thus, $60$ identical packages can be made.

Step 4: Determine the contents of each package

To determine how many of each item is in a packet, divide the total number of each item by the total number of packages ( $60$ ). Each package will contain $3$ chocolates, $4$ pencils, and $5$ erasers.

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