What's the maximum number of Cartons each stack can hold if there are 144 Cartons of Coke and 90 Cartons of Pepsi?

What's the maximum number of Cartons each stack can hold if there are 144 Cartons of Coke and 90 Cartons of Pepsi?

To determine the greatest number of cartons each stack can have, given 144 cartons of coke cans and 90 cartons of pepsi cans need to be stacked separately with each stack having the same number of cartons, use the concept of the Highest Common Factor (HCF) or Greatest Common Divisor (GCD).

We have 144 cartons of coke cans and 90 cartons of pepsi cans. Each type of carton needs to be stacked such that every stack contains the same number of cartons.

We use Euclid's algorithm to find the HCF of 144 and 90. The steps involve repeated division until we reach a remainder of zero:

Divide 144 by 90: 144 = 90 x 1 + 54

Divide 90 by 54: 90 = 54 x 1 + 36

Divide 54 by 36: 54 = 36 x 1 + 18

Divide 36 by 18: 36 = 18 x 2 + 0

The last non-zero remainder in this process is 18

So the HCF of 144 and 90 is 18

Therefore, the greatest number of cartons each stack can have, whether for coke or pepsi cartons, is 18.

What is HCF and GCD?

The concept of Highest Common Factor (HCF) or Greatest Common Divisor (GCD) is fundamental in mathematics, particularly in scenarios involving division and partitioning.

Definition and Purpose:

The HCF/GCD of two or more numbers is the largest number that divides each of them without leaving a remainder. It’s commonly used to determine the largest unit of measure or quantity that can evenly divide given sets of numbers or items.

Euclid's Algorithm:

This ancient algorithm, attributed to Euclid, is widely used to compute the HCF/GCD. It involves a series of division steps where the remainder of one division becomes the divisor for the next step until a remainder of zero is achieved. The last non-zero remainder in this process gives us the HCF/GCD.

Problem Solving Tool:

In competitive exams and problem-solving scenarios, knowledge of the HCF/GCD is essential for quickly determining optimal solutions to numerical problems involving multiple quantities or measurements.

Applications:

In practical applications such as organizing cartons of different items into stacks, as in the example provided, the HCF/GCD helps ensure that each stack contains an equal number of items. This ensures efficient use of space and uniformity in storage. It simplifies fractions by dividing the numerator and denominator by their HCF/GCD. It’s used in scheduling, where it helps determine the smallest interval for repeating events. Engineers use it in designing structures to ensure uniformity in dimensions and materials.