What Is 72 Divisible By

What is 72 Divisible By? A Comprehensive Exploration of Divisibility Rules and Factorization

Understanding divisibility is a fundamental concept in mathematics, crucial for simplifying calculations, solving equations, and grasping more advanced topics. This article delves into the question: "What is 72 divisible by?" We will explore not just the numbers 72 is divisible by, but also the underlying principles of divisibility, different methods for finding factors, and the broader mathematical context. This exploration will be beneficial for students learning about factors and divisibility, as well as anyone interested in refreshing their foundational math skills.

Understanding Divisibility

Divisibility, simply put, refers to whether a number can be divided by another number without leaving a remainder. If a number a is divisible by a number b, it means that a/b results in a whole number (an integer). We often express this as "a is a multiple of b", or "b is a factor of a".

Finding the Factors of 72: A Step-by-Step Approach

Let's systematically find all the numbers that 72 is divisible by. We can achieve this using several methods:

1. Prime Factorization: The Building Blocks

Prime factorization is a powerful technique. It involves breaking down a number into its prime factors – numbers divisible only by 1 and themselves. For 72:

• Start by dividing by the smallest prime number, 2: 72 ÷ 2 = 36
• Continue dividing by 2: 36 ÷ 2 = 18; 18 ÷ 2 = 9
• Now, we encounter a prime number other than 2: 9 is divisible by 3: 9 ÷ 3 = 3
• Finally, we're left with the prime number 3: 3 ÷ 3 = 1

Therefore, the prime factorization of 72 is 2 x 2 x 2 x 3 x 3, or 2³ x 3².

This tells us immediately that 72 is divisible by 2, 3, 4 (2x2), 6 (2x3), 8 (2x2x2), 9 (3x3), 12 (2x2x3), 18 (2x3x3), 24 (2x2x2x3), and 36 (2x2x3x3).

2. Systematic Division: A Hands-On Approach

We can also find the factors of 72 by systematically dividing 72 by each integer starting from 1, checking for remainders:

• 72 ÷ 1 = 72 (Remainder 0)
• 72 ÷ 2 = 36 (Remainder 0)
• 72 ÷ 3 = 24 (Remainder 0)
• 72 ÷ 4 = 18 (Remainder 0)
• 72 ÷ 6 = 12 (Remainder 0)
• 72 ÷ 8 = 9 (Remainder 0)
• 72 ÷ 9 = 8 (Remainder 0)
• 72 ÷ 12 = 6 (Remainder 0)
• 72 ÷ 18 = 4 (Remainder 0)
• 72 ÷ 24 = 3 (Remainder 0)
• 72 ÷ 36 = 2 (Remainder 0)
• 72 ÷ 72 = 1 (Remainder 0)

Notice that after 8, the factors start repeating (9, 8; 12, 6; 18, 4; 24,3; 36,2). This is a characteristic of finding factors.

3. Divisibility Rules: Shortcuts and Insights

Divisibility rules offer quicker ways to check for divisibility without performing long division. Let's apply some to 72:

• Divisibility by 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8). 72 ends in 2, 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. 7 + 2 = 9, which is divisible by 3, so 72 is divisible by 3.
• Divisibility by 4: A number is divisible by 4 if its last two digits are divisible by 4. 72's last two digits (72) are divisible by 4 (72 ÷ 4 = 18), so 72 is divisible by 4.
• Divisibility by 6: A number is divisible by 6 if it's divisible by both 2 and 3. Since 72 is divisible by both, it's divisible by 6.
• Divisibility by 8: A number is divisible by 8 if its last three digits are divisible by 8. Since 72 is only two digits, we need to test directly, which gives 72/8=9, and it's divisible by 8.
• Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9. The sum of digits in 72 is 9, which is divisible by 9, confirming 72's divisibility by 9.
• Divisibility by 12: A number is divisible by 12 if it's divisible by both 3 and 4. Since 72 is divisible by both, it's divisible by 12.

The Complete List of Factors of 72

Combining the results from our different methods, the complete list of factors of 72 is: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.

Understanding Factors and Multiples

It's important to differentiate between factors and multiples. Factors are numbers that divide evenly into a given number, while multiples are numbers that result from multiplying a given number by an integer. In the case of 72:

• Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
• Multiples of 72: 72, 144, 216, 288, 360, and so on.

The Significance of Divisibility in Mathematics

Divisibility is a foundational concept that extends far beyond basic arithmetic. It plays a critical role in:

• Number Theory: Exploring properties of numbers, prime numbers, and their relationships.
• Algebra: Factoring polynomials, simplifying expressions, and solving equations.
• Calculus: Finding limits, derivatives, and integrals.
• Cryptography: Developing secure encryption methods.
• Computer Science: Designing efficient algorithms and data structures.

Frequently Asked Questions (FAQ)

Q: Is 72 a prime number?

A: No, 72 is a composite number because it has factors other than 1 and itself.

Q: How can I quickly check if a large number is divisible by 72?

A: Check if the number is divisible by both 8 and 9. If it is, then it’s divisible by 72 (since 72 = 8 x 9). This combines the divisibility rules for 8 and 9 for a more efficient check.

Q: What are the divisors of 72 that are also prime numbers?

A: The prime divisors of 72 are 2 and 3.

Q: Can a number have an infinite number of factors?

A: No, a number has a finite number of factors.

Q: Is there a formula to find all factors of a number?

A: While there isn't a single formula to directly list all factors, prime factorization provides a systematic approach to find them.

Conclusion: Mastering Divisibility

Understanding what 72 is divisible by – and, more broadly, the concept of divisibility – is crucial for building a strong foundation in mathematics. By combining prime factorization, systematic division, and divisibility rules, we can efficiently determine the factors of any number. This understanding not only helps with basic calculations but also opens doors to more advanced mathematical concepts and applications. Mastering divisibility is a key to unlocking a deeper appreciation of the elegance and power of numbers. Remember, practice is key; the more you work with these concepts, the more intuitive they will become.