There are 75 copies, 100 pens, and 125 pencils in a shop. (a) What is the greatest number of students to distribute stationary goods equally?
Answer (2)
To determine the greatest number of students to whom the shopkeeper can distribute the items equally, we need to find the greatest common divisor (GCD) of the numbers 75, 100, and 125. The GCD is the largest number that can divide each of the numbers without leaving a remainder.
Let's calculate the GCD step by step:
Prime Factorization :
For 75: 75 = 3 × 5 2
For 100: 100 = 2 2 × 5 2
For 125: 125 = 5 3
Calculate GCD :
Identify the common prime factors: The common prime factor is 5.
The smallest power of the common prime factors is 5 2 .
Therefore, the GCD is 5 2 = 25 .
This means the greatest number of students to whom the shopkeeper can distribute each type of item equally is 25. Each student will receive 3 copies, 4 pens, and 5 pencils as calculated here:
25 75 = 3 copies per student.
25 100 = 4 pens per student.
25 125 = 5 pencils per student.
The greatest number of students that can receive stationary goods equally from the shop is 25. Each student will receive 3 copies, 4 pens, and 5 pencils. This was determined by calculating the greatest common divisor of the quantities of each item.
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