Question

There are 44 boys and 32 girls in a class. These students are arranged in rows for a prayer in such a way that each row consists of only either boys or girls, and every row contains an equal number of students. Find the minimum number of rows in which all the students can be arranged. (1) 4 (2) 12 (3) 15 (4) 19

Short answer

Answer: 19 rows.

Step-by-step solution

Find the GCD of boys and girls number

We will find the GCD of the number of boys (44) and girls (32). The GCD can be found using the Euclidean algorithm. We will divide the larger number by the smaller one and keep replacing the larger number by the remainder until we get a remainder of zero.

Euclidean Algorithm

Applying the Euclidean algorithm: Divide 44 by 32: 44 = 32 * 1 + 12 Now replace the larger number (44) by the remainder (12): Divide 32 by 12: 32 = 12 * 2 + 8 Now replace the larger number (32) by the remainder (8): Divide 12 by 8: 12 = 8 * 1 + 4 Now replace the larger number (12) by the remainder (4): Divide 8 by 4: 8 = 4 * 2 + 0 Now we have a remainder of zero. The divisor (4) at this step is our GCD, which represents the number of students per row.

Find the number of rows for boys and girls

Since we found out that the GCD is 4, meaning there will be 4 students per row, now we will find how many rows are required for boys and girls. Number of rows for boys: \(\frac{44}{4} = 11\) Number of rows for girls: \(\frac{32}{4} = 8\)

Add the rows for boys and girls to find the minimum number of rows

Now, to find the minimum number of rows, we simply add the number of rows calculated for both boys and girls: Minimum number of rows: 11 (rows for boys) + 8 (rows for girls) = 19 So, the correct answer is: (4) 19

Key concepts

Euclidean algorithm

The Euclidean algorithm is a time-tested technique for calculating the greatest common divisor (GCD) of two numbers. Infinitely versatile, it's an essential tool in the field of mathematics, particularly number theory. How does it work? In essence, the algorithm involves a succession of divisions where the remainder replaces the smaller number in the next iteration. This process is repeated until the remainder is zero. At this point, the last non-zero remainder is the GCD of the initial two numbers.

For example, consider finding the GCD of 44 and 32. We start by dividing 44 by 32, which gives us a remainder of 12. Next, we divide 32 by 12, leading to a remainder of 8. We continue this pattern: divide 12 by 8, arriving at a remainder of 4, and finally, divide 8 by 4, where the remainder is zero. The GCD, thus, is 4. It's this systematic 'remainder replacement' strategy that makes the Euclidean algorithm both powerful and reliable in finding the greatest common divisor.

Arranging students in rows

When tasked with organizing students into rows, whether for a class photo or a school assembly, you're putting into practice a fundamental concept in discrete mathematics. The goal is to achieve uniformity and order, with each row containing an equal number of students. To find the optimal arrangement, the idea is to divide the students into groups, with the size of each group being the GCD of the total number of students in the subsets - in our case, boys and girls.

Using our example of 44 boys and 32 girls, and the GCD of 4, we can organize them into rows with 4 students each. While it might not be immediately obvious, such arrangements are not just about symmetry; they are closely tied to factors like maximizing interaction, ensuring everyone is visible, and can also be a subtle exercise in resource allocation. It's a practical application of GCD in a way that touches everyday life - from classroom settings to organizing teams in workplaces.

Math problem solving

Solving math problems, like arranging students in rows, often requires a structured approach. Clarity of the question is vital, understanding what is being asked – in our case, the minimum number of rows to arrange students. It's beneficial to break down the problem into manageable steps, first identifying the right method, such as finding the GCD using the Euclidean algorithm in this scenario. Then, applying the concept to come to a solution.

By systematically working through these steps—identifying the GCD as the size of each group (or row) and dividing the total number of each subset by the GCD—we neatly wrap up the problem. This not only ensures that the solution is efficient and correct but also reinforces a logical and methodical mindset that's crucial in the realm of mathematics and beyond. Developing this problem-solving skillset is indispensable for students as they tackle diverse challenges throughout different areas of study and life.