What Is 90 Divisible By

What is 90 Divisible By? Unlocking the World of Divisibility Rules

Understanding divisibility is a fundamental concept in mathematics, crucial for simplifying calculations and solving various problems. This article delves into the divisibility of 90, exploring the numbers that divide 90 evenly without leaving a remainder. We'll examine the divisibility rules, explore the prime factorization of 90, and uncover the significance of understanding divisibility in broader mathematical contexts. This comprehensive guide will empower you to confidently determine the divisors of any number, not just 90.

Understanding Divisibility

Before we dive into the specifics of 90, let's clarify what divisibility means. A number is divisible by another number if the result of their division is a whole number (an integer) with no remainder. For example, 10 is divisible by 5 because 10 ÷ 5 = 2, a whole number. However, 10 is not divisible by 3 because 10 ÷ 3 = 3 with a remainder of 1.

Finding the Divisors of 90: A Step-by-Step Approach

There are several methods to determine all the numbers that 90 is divisible by. Let's explore these methods systematically:

1. Listing Factors

The most straightforward approach is to systematically list the factors of 90. Factors are numbers that divide evenly into 90. We can start with 1 and proceed until we reach 90, checking each number:

• 1: 90 ÷ 1 = 90
• 2: 90 ÷ 2 = 45
• 3: 90 ÷ 3 = 30
• 5: 90 ÷ 5 = 18
• 6: 90 ÷ 6 = 15
• 9: 90 ÷ 9 = 10
• 10: 90 ÷ 10 = 9
• 15: 90 ÷ 15 = 6
• 18: 90 ÷ 18 = 5
• 30: 90 ÷ 30 = 3
• 45: 90 ÷ 45 = 2
• 90: 90 ÷ 90 = 1

Therefore, the divisors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90.

2. Prime Factorization

Prime factorization is a powerful technique for determining all the divisors of a number. It involves breaking down the number into its prime factors—numbers divisible only by 1 and themselves. Let's find the prime factorization of 90:

• 90 can be divided by 2: 90 = 2 × 45
• 45 can be divided by 3: 45 = 3 × 15
• 15 can be divided by 3: 15 = 3 × 5
• 5 is a prime number.

Therefore, the prime factorization of 90 is 2 × 3 × 3 × 5, or 2¹ × 3² × 5¹.

Once you have the prime factorization, you can find all the divisors by systematically combining the prime factors. For example:

• 2¹ = 2
• 3¹ = 3
• 5¹ = 5
• 2¹ × 3¹ = 6
• 2¹ × 5¹ = 10
• 3¹ × 3¹ = 9
• 3¹ × 5¹ = 15
• 2¹ × 3¹ × 3¹ = 18
• 2¹ × 3¹ × 5¹ = 30
• 3¹ × 3¹ × 5¹ = 45
• 2¹ × 3¹ × 3¹ × 5¹ = 90
• 1 (The product of no prime factors)

This method systematically generates all the divisors, ensuring none are missed.

3. Divisibility Rules

Divisibility rules provide shortcuts for determining if a number is divisible by specific numbers without performing the division. Let's look at some relevant divisibility rules and how they apply to 90:

• Divisibility by 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8). Since 90 ends in 0, it's divisible by 2.
• Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3. The sum of the digits in 90 (9 + 0 = 9) is divisible by 3, so 90 is divisible by 3.
• Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5. Since 90 ends in 0, it's divisible by 5.
• Divisibility by 6: A number is divisible by 6 if it's divisible by both 2 and 3. Since 90 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 in 90 (9 + 0 = 9) is divisible by 9, so 90 is divisible by 9.
• Divisibility by 10: A number is divisible by 10 if its last digit is 0. Since 90 ends in 0, it's divisible by 10.
• Divisibility by 15: A number is divisible by 15 if it's divisible by both 3 and 5. Since 90 is divisible by both 3 and 5, it is divisible by 15.

By applying these rules, we quickly confirm that 90 is divisible by 2, 3, 5, 6, 9, 10, and 15. We can then use these factors to find other divisors, such as 18 (2 x 9), 30 (2 x 15), 45 (3 x 15), and so on, systematically reaching all the divisors as we did in the first method.

The Significance of Divisibility

Understanding divisibility isn't just about finding factors; it's a fundamental concept with wide-ranging applications in mathematics and beyond:

• Simplification of Fractions: Divisibility helps simplify fractions to their lowest terms by finding the greatest common divisor (GCD) of the numerator and denominator.
• Solving Equations: Divisibility can help determine possible solutions to equations involving integers.
• Number Theory: Divisibility is central to number theory, a branch of mathematics that explores the properties of integers. Concepts like prime numbers, relatively prime numbers, and modular arithmetic all rely heavily on divisibility.
• Computer Science: Divisibility plays a role in algorithms and data structures in computer science, particularly in areas like cryptography and sorting algorithms.
• Real-World Applications: Divisibility is used in everyday life for tasks such as dividing resources evenly, determining the number of items that fit into containers, and calculating proportions.

Frequently Asked Questions (FAQs)

Q: Is 90 divisible by 4?

A: No, 90 is not divisible by 4. A quick check: 90 ÷ 4 = 22 with a remainder of 2.

Q: Is 90 a perfect number?

A: No, 90 is not a perfect number. A perfect number is a positive integer that is equal to the sum of its proper divisors (divisors excluding the number itself). The sum of the proper divisors of 90 (1 + 2 + 3 + 5 + 6 + 9 + 10 + 15 + 18 + 30 + 45 = 144) is greater than 90.

Q: What is the greatest common divisor (GCD) of 90 and 120?

A: To find the GCD, we can use prime factorization. The prime factorization of 90 is 2 × 3² × 5, and the prime factorization of 120 is 2³ × 3 × 5. The common factors are 2¹, 3¹, and 5¹. Therefore, the GCD of 90 and 120 is 2 × 3 × 5 = 30.

Q: How many divisors does 90 have?

A: As we determined earlier, 90 has 12 divisors: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90.

Q: How can I find the divisors of larger numbers efficiently?

A: For larger numbers, prime factorization combined with systematic combination of prime factors is the most efficient approach. Computer programs can greatly assist in finding the prime factorization and subsequent divisors of very large numbers.

Conclusion

This exploration of the divisibility of 90 illustrates the fundamental importance of understanding divisibility rules and prime factorization. These mathematical tools are not just for solving textbook problems; they are essential for simplifying calculations, solving more complex mathematical problems, and tackling real-world applications requiring precise calculations and divisions. By mastering these concepts, you'll build a stronger foundation in mathematics and open doors to further exploration of more advanced mathematical concepts. Remember to practice regularly – the more you work with these concepts, the more intuitive and efficient your approach will become.