What Is 63 Divisible By

What is 63 Divisible By? Unlocking the Secrets of Divisibility

This article delves into the fascinating world of divisibility, specifically exploring all the numbers by which 63 is divisible. Understanding divisibility rules is crucial for simplifying calculations, solving mathematical problems, and developing a stronger foundation in number theory. We'll explore the concept of divisibility, outline the divisibility rules, determine all the divisors of 63, and even touch upon the prime factorization, providing a comprehensive understanding of this seemingly simple yet insightful mathematical concept.

Understanding Divisibility

Divisibility, in simple terms, refers to whether a number can be divided by another number without leaving a remainder. If a number a is divisible by another number b, it means that the division of a by b results in a whole number (an integer). In other words, there's no fractional part in the quotient. For example, 12 is divisible by 3 because 12 ÷ 3 = 4, a whole number. However, 12 is not divisible by 5 because 12 ÷ 5 = 2 with a remainder of 2.

The numbers that divide a given number without leaving a remainder are called its divisors or factors. Finding all the divisors of a number is a fundamental concept in number theory and has applications in various mathematical fields.

Divisibility Rules: Your Shortcuts to Success

Before we determine what numbers divide 63, let's review some essential divisibility rules. These rules provide quick ways to check for divisibility without performing long division. These shortcuts are particularly useful when dealing with larger numbers.

• Divisibility by 1: Every integer is divisible by 1.

• Divisibility by 2: A number is divisible by 2 if its last digit is an even number (0, 2, 4, 6, or 8).

• Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.

• Divisibility by 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.

• Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5.

• Divisibility by 6: A number is divisible by 6 if it is divisible by both 2 and 3.

• Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.

• Divisibility by 10: A number is divisible by 10 if its last digit is 0.

Finding the Divisors of 63: A Step-by-Step Approach

Now, let's apply these rules and other methods to find all the numbers that divide 63 without leaving a remainder.

1. Start with the obvious: We know that every number is divisible by 1 and itself. Therefore, 1 and 63 are divisors of 63.

2. Apply the divisibility rules:

   • Divisibility by 2: 63 is not divisible by 2 because its last digit (3) is odd.
   • Divisibility by 3: The sum of the digits of 63 (6 + 3 = 9) is divisible by 3, so 63 is divisible by 3.
   • Divisibility by 4: The last two digits of 63 are 63, which is not divisible by 4, so 63 is not divisible by 4.
   • Divisibility by 5: The last digit of 63 is 3, which is not 0 or 5, so 63 is not divisible by 5.
   • Divisibility by 6: Since 63 is divisible by 3 but not by 2, it's not divisible by 6.
   • Divisibility by 9: The sum of the digits (9) is divisible by 9, so 63 is divisible by 9.
   • Divisibility by 10: The last digit is not 0, so 63 is not divisible by 10.

3. Check for other divisors: We've found that 1, 3, 9, and 63 are divisors. To ensure we haven't missed any, we can systematically check numbers between 1 and 63. However, since we know 63 = 9 x 7, we already have most of the factors. Considering that 7 is a prime number, we can ascertain that it's also a divisor of 63.

4. Listing all the divisors: Therefore, the complete list of divisors of 63 is: 1, 3, 7, 9, 21, and 63.

Prime Factorization of 63: A Deeper Dive

Prime factorization is the process of expressing a number as a product of its prime factors (numbers only divisible by 1 and themselves). Understanding the prime factorization of a number reveals important information about its divisors.

The prime factorization of 63 is 3² x 7. This means that 63 can be expressed as 3 x 3 x 7. This factorization helps us understand why 3, 7, 9 (3x3), and 21 (3x7) are divisors of 63. Every combination of these prime factors will result in a divisor.

Frequently Asked Questions (FAQ)

Q: How can I quickly determine if a number is divisible by 7? There isn't a simple divisibility rule for 7 like there is for 3 or 9. The most straightforward method is to perform the division.

Q: Are there any tricks to finding all divisors of a larger number? For larger numbers, prime factorization becomes increasingly important. Once you have the prime factorization, you can systematically find all possible combinations of the prime factors to obtain all divisors.

Q: What is the significance of understanding divisibility? Understanding divisibility is fundamental in simplifying calculations, solving various mathematical problems, and grasping more advanced concepts in number theory and algebra.

Conclusion: Mastering Divisibility, One Number at a Time

We've explored the divisibility of 63 comprehensively, determining that its divisors are 1, 3, 7, 9, 21, and 63. We've also examined the concept of divisibility, reviewed helpful divisibility rules, and explored the prime factorization of 63 (3² x 7). This journey into the world of divisibility illustrates how seemingly simple mathematical concepts can provide a rich understanding of numbers and their relationships. By mastering divisibility rules and the concept of prime factorization, you'll significantly enhance your mathematical skills and problem-solving abilities. Remember, the key is practice! The more you work with numbers and explore their properties, the more intuitive and effortless these concepts will become.