What Is 108 Divisible By

What is 108 Divisible By? Unlocking the Secrets of Divisibility

The seemingly simple question, "What is 108 divisible by?" opens a fascinating door into the world of number theory and divisibility rules. Understanding divisibility isn't just about finding factors; it's about grasping fundamental mathematical relationships and developing efficient problem-solving skills. This article will delve deep into the divisors of 108, exploring various methods to identify them, explaining the underlying mathematical principles, and even touching upon the number's significance in different cultures and contexts.

Understanding Divisibility

Before we tackle 108 specifically, let's establish a foundational understanding of divisibility. A number is divisible by another number if the division results in a whole number (no remainder). For example, 12 is divisible by 3 because 12/3 = 4. Conversely, 12 is not divisible by 5 because 12/5 = 2 with a remainder of 2.

Finding the Divisors of 108: A Step-by-Step Approach

There are several ways to find all the numbers that 108 is divisible by. Let's explore the most common and effective methods:

1. Prime Factorization:

This is arguably the most fundamental and powerful technique. Prime factorization involves breaking down a number into its prime factors – numbers divisible only by 1 and themselves (e.g., 2, 3, 5, 7, 11...). Let's find the prime factorization of 108:

• 108 is an even number, so it's divisible by 2: 108 ÷ 2 = 54
• 54 is also even: 54 ÷ 2 = 27
• 27 is divisible by 3: 27 ÷ 3 = 9
• 9 is divisible by 3: 9 ÷ 3 = 3
• 3 is a prime number.

Therefore, the prime factorization of 108 is 2 x 2 x 3 x 3 x 3, or 2² x 3³.

Knowing the prime factorization is crucial because it allows us to easily identify all divisors. To find the divisors, we systematically combine the prime factors:

• Combinations of 2: 1, 2, 4 (1, 2², 2)
• Combinations of 3: 1, 3, 9, 27 (3⁰, 3¹, 3², 3³)
• Combinations of 2 and 3: Combining the factors of 2 and 3 gives us additional divisors: 6, 12, 18, 36, 54

Therefore, the complete list of divisors of 108 is: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108.

2. Using Divisibility Rules:

Divisibility rules provide quick checks for certain divisors without performing the actual division. Let's see how they apply to 108:

• Divisibility by 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8). 108 ends in 8, so it's divisible by 2.
• Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3. 1 + 0 + 8 = 9, and 9 is divisible by 3, so 108 is divisible by 3.
• Divisibility by 4: A number is divisible by 4 if its last two digits are divisible by 4. 08 is divisible by 4, so 108 is divisible by 4.
• Divisibility by 6: A number is divisible by 6 if it's divisible by both 2 and 3. Since 108 is divisible by both 2 and 3, it's divisible by 6.
• Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9. The sum of the digits of 108 is 9, which is divisible by 9, so 108 is divisible by 9.
• Divisibility by 12: A number is divisible by 12 if it's divisible by both 3 and 4. Since 108 is divisible by both 3 and 4, it is divisible by 12.

These rules help quickly identify several divisors, but prime factorization provides a more comprehensive and systematic approach.

3. Systematic Division:

This is a more brute-force method, but it can be useful for smaller numbers. You systematically divide the number by integers, starting from 1 and increasing until you reach the square root of the number. Any integer that divides 108 without a remainder is a divisor. The divisors found this way will be less than or equal to the square root of 108, approximately 10.39. After finding these, their corresponding pairs can be identified. For instance, if 2 is a divisor, then 108/2 = 54 is also a divisor.

Mathematical Properties and Significance

Beyond simply identifying its divisors, 108 possesses several interesting mathematical properties and cultural significance:

• Abundant Number: 108 is an abundant number because the sum of its proper divisors (divisors excluding itself) is greater than 108. The sum of its proper divisors is 216.
• Highly Composite Number: Although not strictly a highly composite number (which have more divisors than any smaller positive integer), it has a relatively large number of divisors for its size.
• Spiritual Significance: The number 108 holds significant meaning in various spiritual traditions, including Hinduism, Buddhism, and Jainism. It's often associated with the number of beads in a mala (prayer beads) and is believed to represent spiritual completeness. Its connection to sacred geometry and spiritual practices adds another layer of interest to this number.
• Geometric Significance: The number 108 appears in various geometric contexts, such as the relationship between the radius and the circumference of a circle. The ratio of the circumference to the diameter of a circle is π (pi), approximately 3.14159. While not directly related to divisibility, it illustrates the unexpected connections found in mathematics.

Frequently Asked Questions (FAQ)

• Q: Is 108 a perfect number?

  • A: No. A perfect number is a positive integer that is equal to the sum of its proper divisors. The sum of 108's proper divisors (excluding 108 itself) is 216, not 108.

• Q: What is the greatest common divisor (GCD) of 108 and another number, say 72?

  • A: To find the GCD, we can use prime factorization. The prime factorization of 72 is 2³ x 3². Comparing the prime factorizations of 108 (2² x 3³) and 72 (2³ x 3²), the common factors are 2² and 3². Therefore, the GCD of 108 and 72 is 2² x 3² = 4 x 9 = 36.

• Q: How many divisors does 108 have?

  • A: 108 has 12 divisors: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108.

• Q: What is the least common multiple (LCM) of 108 and another number, say 72?

  • A: The LCM is the smallest positive integer that is divisible by both numbers. Using the prime factorizations, the LCM of 108 (2² x 3³) and 72 (2³ x 3²) is 2³ x 3³ = 8 x 27 = 216.

Conclusion

The seemingly simple question of what 108 is divisible by leads us on a journey through the fascinating world of number theory. By understanding prime factorization, divisibility rules, and systematic approaches, we can efficiently identify all the divisors of 108 and appreciate its unique mathematical and cultural significance. This exploration extends beyond mere calculations; it highlights the interconnectedness of mathematical concepts and reveals the beauty and depth inherent in seemingly simple numbers. The number 108, with its rich tapestry of mathematical properties and cultural symbolism, serves as a perfect example of how a single number can hold a multitude of meanings and fascinate mathematicians, spiritual seekers, and curious minds alike.