The Number of Fruits in the Largest Heap

Oct 13, 2025 by talynt.org 2 views

Introduction

In this intriguing problem, we are presented with a scenario where Hamida has a vast collection of fruits, consisting of apples, chickoos, and oranges. The task at hand is to distribute these fruits into equal heaps, with the ultimate goal of determining the number of fruits in the largest heap and the number of such heaps that will be formed. This problem requires a deep understanding of mathematical concepts, particularly those related to divisibility and the greatest common divisor (GCD).

The Problem Statement

Hamida has 527 apples, 646 chickoos, and 748 oranges. These fruits have to be distributed in equal heaps. What would be the number of fruits in the largest heap? How many such heaps will be formed?

Breaking Down the Problem

To tackle this problem, we need to find the greatest common divisor (GCD) of the three numbers: 527, 646, and 748. The GCD is the largest number that divides all three numbers without leaving a remainder. This is a crucial step, as it will help us determine the number of fruits in the largest heap.

Finding the Greatest Common Divisor (GCD)

To find the GCD of 527, 646, and 748, we can use the Euclidean algorithm. This algorithm involves a series of steps to find the GCD of two numbers, and then we can extend it to find the GCD of three numbers.

Step 1: Find the GCD of 527 and 646

To find the GCD of 527 and 646, we can use the Euclidean algorithm:

527 = 646 × 0 + 527 646 = 527 × 1 + 119 527 = 119 × 4 + 27 119 = 27 × 4 + 11 27 = 11 × 2 + 5 11 = 5 × 2 + 1 5 = 1 × 5 + 0

Therefore, the GCD of 527 and 646 is 1.

Step 2: Find the GCD of 527 and 748

To find the GCD of 527 and 748, we can use the Euclidean algorithm:

748 = 527 × 1 + 221 527 = 221 × 2 + 85 221 = 85 × 2 + 51 85 = 51 × 1 + 34 51 = 34 × 1 + 17 34 = 17 × 2 + 0

Therefore, the GCD of 527 and 748 is 17.

Step 3: Find the GCD of 646 and 748

To find the GCD of 646 and 748, we can use the Euclidean algorithm:

748 = 646 × 1 + 102 646 = 102 × 6 + 34 102 = 34 × 3 + 0

Therefore, the GCD of 646 and 748 is 34.

Step 4: Find the GCD of 527, 646, and 748

Now that we have found the GCD of each pair of numbers, we can find the GCD of all three numbers. Since the GCD of 527 and 646 is 1, and the GCD of 527 and 748 is 17, and the GCD of 646 and 748 is 34, we can conclude that the GCD of 527, 646, and 748 is 1.

The Number of Fruits in the Largest Heap

Since the GCD of 527, 646, and 748 is 1, the number of fruits in the largest heap will be the largest of the three numbers, which is 748.

The Number of Heaps

To find the number of heaps, we can divide the total number of fruits by the number of fruits in the largest heap. In this case, we have:

Total number of fruits = 527 + 646 + 748 = 1921 Number of fruits in the largest heap = 748

Therefore, the number of heaps will be:

1921 ÷ 748 = 2.57

Since we cannot have a fraction of a heap, we will round down to the nearest whole number. Therefore, the number of heaps will be 2.

Conclusion

In conclusion, the number of fruits in the largest heap is 748, and the number of heaps will be 2. This problem required a deep understanding of mathematical concepts, particularly those related to divisibility and the greatest common divisor (GCD). By breaking down the problem into smaller steps and using the Euclidean algorithm, we were able to find the GCD of the three numbers and determine the number of fruits in the largest heap and the number of heaps that will be formed.

References

Euclidean algorithm
Greatest common divisor (GCD)
Divisibility
Mathematical concepts

Q: What is the greatest common divisor (GCD) and why is it important in this problem?

A: The greatest common divisor (GCD) is the largest number that divides all three numbers (527, 646, and 748) without leaving a remainder. It is important in this problem because it helps us determine the number of fruits in the largest heap.

Q: How do you find the GCD of three numbers?

A: To find the GCD of three numbers, we can use the Euclidean algorithm. This algorithm involves a series of steps to find the GCD of two numbers, and then we can extend it to find the GCD of three numbers.

Q: What is the Euclidean algorithm and how does it work?

A: The Euclidean algorithm is a method for finding the greatest common divisor (GCD) of two numbers. It works by repeatedly dividing the larger number by the smaller number and taking the remainder. This process is repeated until the remainder is zero, at which point the GCD is the last non-zero remainder.

Q: Why did we get a GCD of 1 for the three numbers?

A: We got a GCD of 1 because the three numbers (527, 646, and 748) are relatively prime, meaning they have no common factors other than 1.

Q: How do you determine the number of fruits in the largest heap?

A: To determine the number of fruits in the largest heap, we divide the total number of fruits by the GCD of the three numbers. In this case, we have:

Total number of fruits = 527 + 646 + 748 = 1921 GCD of the three numbers = 1

Therefore, the number of fruits in the largest heap is 1921.

Q: How do you determine the number of heaps?

A: To determine the number of heaps, we divide the total number of fruits by the number of fruits in the largest heap. In this case, we have:

Total number of fruits = 1921 Number of fruits in the largest heap = 748

Therefore, the number of heaps is 1921 ÷ 748 = 2.57, which we round down to 2.

Q: What if the GCD of the three numbers was not 1?

A: If the GCD of the three numbers was not 1, we would need to divide the total number of fruits by the GCD to determine the number of fruits in the largest heap. We would then divide the total number of fruits by the number of fruits in the largest heap to determine the number of heaps.

Q: Can you provide an example of a different set of numbers that would result in a different number of heaps?

A: Yes, let's consider the following set of numbers:

120 apples
180 chickoos
240 oranges

To find the GCD of these numbers, we can use the Euclidean algorithm:

120 = 180 × 0 + 120 180 = 120 × 1 + 60 120 = 60 × 2 + 0

Therefore, the GCD of 120, 180, and 240 is 60.

To determine the number of fruits in the largest heap, we divide the total number of fruits by the GCD:

Total number of fruits = 120 + 180 + 240 = 540 GCD of the three numbers = 60

Therefore, the number of fruits in the largest heap is 540 ÷ 60 = 9.

To determine the number of heaps, we divide the total number of fruits by the number of fruits in the largest heap:

Total number of fruits = 540 Number of fruits in the largest heap = 9

Therefore, the number of heaps is 540 ÷ 9 = 60.

Conclusion

In conclusion, the great fruit distribution problem requires a deep understanding of mathematical concepts, particularly those related to divisibility and the greatest common divisor (GCD). By breaking down the problem into smaller steps and using the Euclidean algorithm, we were able to find the GCD of the three numbers and determine the number of fruits in the largest heap and the number of heaps that will be formed.

References

Euclidean algorithm
Greatest common divisor (GCD)
Divisibility
Mathematical concepts