Express each number as a product of its prime factors: 84 = 2 × 2 × 3 × 7 = 2² × 3 × 7, and 126 = 2 × 3 × 3 × 7 = 2 × 3² × 7.

Divide the total number of red pens by 42 to find red pens per student: 84 ÷ 42 = 2. Divide the total number of blue pens by 42 to find blue pens per student: 126 ÷ 42 = 3.

a teacher has 84 red pens and 126 blue pens he wants to distribute them equally among the maximum possible number of students without any pain left over what is the maximum number of students how many red pens will eat student get how many blue pens will eat student get

Answer: The maximum number of students is 42. Each student will get 2 red pens and 3 blue pens.

Step-by-step solution

Step 1: Identify the required mathematical operation

To divide both $84$ red pens and $126$ blue pens equally among students without any remaining pens, the number of students must divide both $84$ and $126$ exactly. To maximize this count, we need to find the Highest Common Factor (HCF) of $84$ and $126$ .

Step 2: Find the prime factorisation of 84 and 126

Express each number as a product of its prime factors: $84 = 2 \times 2 \times 3 \times 7 = 2^2 \times 3 \times 7$ , and $126 = 2 \times 3 \times 3 \times 7 = 2 \times 3^2 \times 7$ .

Step 3: Calculate the HCF to determine the maximum students

The $\operatorname{HCF}$ is the product of the smallest power of each common prime factor. The common prime factors are $2$ , $3$ , and $7$ , with minimum powers $2^1$ , $3^1$ , and $7^1$ . Multiplying these gives $2 \times 3 \times 7 = 42$ . Thus, the maximum number of students is $42$ .

Step 4: Calculate the number of pens each student receives

Divide the total number of red pens by $42$ to find red pens per student: $84 \div 42 = 2$ . Divide the total number of blue pens by $42$ to find blue pens per student: $126 \div 42 = 3$ .

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