Assignment 4: divisible by three – Embark on an intriguing journey into the realm of divisibility by three, a mathematical concept that unveils hidden patterns and simplifies calculations across diverse fields. From ancient civilizations to modern-day applications, this exploration promises to captivate your curiosity and enhance your understanding.

Delve into the mathematical foundations of divisibility by three, unraveling the rules and properties that govern this fascinating concept. Discover practical applications in computer science, engineering, and finance, where divisibility by three streamlines problem-solving and enhances efficiency.

Mathematical Properties of Divisibility by Three: Assignment 4: Divisible By Three

In mathematics, divisibility refers to the relationship between two integers. When one integer (the dividend) is evenly divisible by another integer (the divisor), the quotient is a whole number without any remainder.

One of the fundamental properties of divisibility is divisibility by three. A number is divisible by three if the sum of its digits is divisible by three. For example, the number 123 is divisible by three because 1 + 2 + 3 = 6, which is divisible by three.

Mathematical Rule for Divisibility by Three

The mathematical rule for determining if a number is divisible by three is based on the sum of its digits. Specifically, a number is divisible by three if the sum of its digits is divisible by three. This rule can be applied to any integer, regardless of its size or number of digits.

For example, to determine if the number 456 is divisible by three, we add its digits: 4 + 5 + 6 = 15. Since 15 is divisible by three, we can conclude that 456 is also divisible by three.

Applications of Divisibility by Three

Divisibility by three has practical applications in various fields, simplifying calculations and problem-solving.

Computer Science

In computer science, divisibility by three is used for:

• Hashing algorithms: Dividing the hash value by three helps distribute data evenly across buckets, reducing collisions.
• Checksums: Checking if a checksum is divisible by three ensures data integrity during transmission.
• Error detection: In certain error detection codes, divisibility by three helps identify errors in data transmission.

Engineering

In engineering, divisibility by three is used for:

• Load balancing: Dividing a load evenly among three or more components ensures optimal performance.
• Structural design: Checking if the number of bolts or rivets used in a structure is divisible by three ensures proper load distribution.

li>Circuit design: In certain electronic circuits, divisibility by three helps simplify calculations and reduce component count.

Finance

In finance, divisibility by three is used for:

• Investment diversification: Dividing investments into three or more asset classes helps reduce risk and balance returns.
• Debt repayment: Checking if a loan amount or repayment schedule is divisible by three helps ensure manageable monthly payments.
• Financial analysis: Dividing certain financial ratios by three simplifies interpretation and comparison across different companies.

Methods for Checking Divisibility by Three

Determining if a number is divisible by three is a fundamental skill in mathematics. There are several methods to check divisibility by three, each with its own advantages and applications.

Divisibility Test Based on the Sum of Digits

This test involves adding the individual digits of a number and checking if the result is divisible by three. If the sum is divisible by three, then the original number is also divisible by three.

• Example:Consider the number 123. The sum of its digits is 1 + 2 + 3 = 6, which is divisible by three. Therefore, 123 is divisible by three.

Divisibility Test Based on the Remainder of Division by Three

This test checks if the remainder of a number when divided by three is zero. If the remainder is zero, then the number is divisible by three.

• Example:Divide 27 by 3. The remainder is 0. Therefore, 27 is divisible by three.

Properties of Numbers Divisible by Three

Numbers that are divisible by three share certain unique properties that distinguish them from other numbers. These properties are closely linked to the fundamental rules of divisibility and provide valuable insights into the behavior of numbers.

Sum of Digits

One of the most notable properties of numbers divisible by three is that the sum of their individual digits is always divisible by three. For instance, the number 123 is divisible by three because the sum of its digits (1 + 2 + 3) is 6, which is also divisible by three.

Alternating Sum of Digits, Assignment 4: divisible by three

Another interesting property involves the alternating sum of digits. If we take the sum of the digits in the odd positions and subtract the sum of the digits in the even positions, the resulting value is always divisible by three.

Consider the number 456: the alternating sum of digits is (4 – 5 + 6) = 5, which is divisible by three.

Relationship with Other Divisibility Rules

The divisibility rule for three is closely related to other divisibility rules. For example, a number is divisible by both three and nine if it is divisible by nine. Additionally, a number is divisible by three if it is divisible by both three and six.

Examples

• The number 36 is divisible by three because its sum of digits (3 + 6) is 9, which is divisible by three.
• The number 153 is divisible by three because its alternating sum of digits (1 – 5 + 3) is 3, which is divisible by three.
• The number 270 is divisible by both three and nine because its sum of digits (2 + 7 + 0) is 9, which is divisible by both three and nine.

Examples and Non-Examples

In mathematics, divisibility is a fundamental concept that determines whether one number is evenly divisible by another. When a number is divisible by three, it means that it can be divided by three without leaving a remainder. Understanding the properties and methods of divisibility by three is essential for various mathematical operations and applications.

To illustrate the concept of divisibility by three, let’s explore examples and non-examples of numbers that fall into these categories:

Examples of Numbers Divisible by Three

• 9: 9 ÷ 3 = 3, leaving no remainder
• 12: 12 ÷ 3 = 4, leaving no remainder
• 18: 18 ÷ 3 = 6, leaving no remainder
• 21: 21 ÷ 3 = 7, leaving no remainder
• 24: 24 ÷ 3 = 8, leaving no remainder

Non-Examples of Numbers Not Divisible by Three

• 5: 5 ÷ 3 = 1 remainder 2
• 10: 10 ÷ 3 = 3 remainder 1
• 14: 14 ÷ 3 = 4 remainder 2
• 17: 17 ÷ 3 = 5 remainder 2
• 22: 22 ÷ 3 = 7 remainder 1

The key distinction between divisible and non-divisible numbers is the presence or absence of a remainder when the number is divided by three. If the remainder is zero, the number is divisible by three; if the remainder is not zero, the number is not divisible by three.

Historical and Cultural Significance

The concept of divisibility by three has played a significant role in various ancient civilizations and religious practices throughout history. Its presence can be traced back to ancient Egypt, where the number three was considered sacred and associated with the gods.

In the Pythagorean tradition, numbers divisible by three were believed to possess mystical properties and were considered perfect.

Role in Ancient Civilizations

• In ancient Egypt, the number three was associated with the gods and was considered a sacred number. The pyramids were built with a base length that was divisible by three, and the number of steps on each side was also divisible by three.

• In ancient Greece, the number three was considered to be a perfect number. The Pythagoreans believed that the universe was governed by mathematical principles, and they associated the number three with the three elements of earth, air, and water.
• In ancient China, the number three was considered to be a lucky number. It was often used in religious ceremonies and rituals, and it was believed to bring good fortune.

Role in Religious Practices

• In Christianity, the number three is associated with the Holy Trinity (Father, Son, and Holy Spirit). It is also found in the three wise men who visited Jesus at his birth and the three days he spent in the tomb before his resurrection.

• In Islam, the number three is associated with the three main pillars of faith: belief in Allah, prayer, and fasting. It is also found in the three holiest cities of Islam: Mecca, Medina, and Jerusalem.
• In Hinduism, the number three is associated with the three main gods: Brahma, Vishnu, and Shiva. It is also found in the three main stages of life: childhood, adulthood, and old age.

Interesting Anecdotes and Stories

• In the Bible, the number three is mentioned over 400 times. One famous example is the story of the three wise men who visited Jesus at his birth. The three wise men brought three gifts: gold, frankincense, and myrrh.

• In the fairy tale “The Three Little Pigs,” the three pigs build three houses: one of straw, one of sticks, and one of bricks. The wolf blows down the first two houses, but he cannot blow down the house of bricks.

  This story teaches the importance of building a strong foundation.

• In the popular song “Three Blind Mice,” three blind mice are chased by a farmer’s wife. The song is often sung to children as a nursery rhyme.

FAQ

What is the divisibility rule for three?

A number is divisible by three if the sum of its digits is divisible by three.

How is divisibility by three used in real-life applications?

Divisibility by three is used in various fields, including computer science (error detection and correction), engineering (load balancing), and finance (check digit verification).

What are some examples of numbers divisible by three?

12, 18, 21, 27, 30, 33, 36, 39, 42, 45