What Is 87 Divisible By

What is 87 Divisible By? Unlocking the Secrets of Divisibility Rules

Finding out what numbers 87 is divisible by might seem like a simple arithmetic problem, but it opens a door to understanding fundamental concepts in number theory and divisibility rules. This exploration goes beyond simply finding the divisors; we'll delve into the why behind the process, exploring efficient methods and solidifying your understanding of number properties. This article will provide a comprehensive guide, suitable for learners of all levels, from elementary school students to those refreshing their mathematical skills.

Introduction: Understanding Divisibility

Divisibility refers to the ability of a number to be divided evenly by another number without leaving a remainder. For example, 12 is divisible by 3 because 12 ÷ 3 = 4 with no remainder. Conversely, 13 is not divisible by 3 because dividing 13 by 3 results in a remainder of 1. Determining divisibility is crucial in simplifying calculations, factoring numbers, and understanding more advanced mathematical concepts.

Our focus today is on the number 87. We will systematically determine all its divisors, explaining the logic behind each step and introducing you to helpful divisibility rules.

Method 1: The Brute Force Approach (Trial Division)

The most straightforward method, although not the most efficient for larger numbers, is trial division. We systematically check each integer starting from 1, seeing if it divides 87 without leaving a remainder.

• 1: 87 ÷ 1 = 87 (87 is divisible by 1. Every integer is divisible by 1.)
• 2: 87 ÷ 2 = 43.5 (87 is not divisible by 2. Even numbers are divisible by 2.)
• 3: 87 ÷ 3 = 29 (87 is divisible by 3. We'll explore the divisibility rule for 3 later.)
• 4: 87 ÷ 4 = 21.75 (87 is not divisible by 4. Numbers divisible by 4 are divisible by 2 twice.)
• 5: 87 ÷ 5 = 17.4 (87 is not divisible by 5. Numbers ending in 0 or 5 are divisible by 5.)
• 6: 87 ÷ 6 = 14.5 (87 is not divisible by 6. Numbers divisible by 6 are divisible by both 2 and 3.)
• 7: 87 ÷ 7 ≈ 12.43 (87 is not divisible by 7.)
• 8: 87 ÷ 8 = 10.875 (87 is not divisible by 8.)
• 9: 87 ÷ 9 ≈ 9.67 (87 is not divisible by 9.)
• 10: 87 ÷ 10 = 8.7 (87 is not divisible by 10.)
• 29: 87 ÷ 29 = 3 (87 is divisible by 29.)
• Beyond 29: Once we reach 29, we’ve found all the divisors. This is because any divisor greater than √87 (approximately 9.3) will have a corresponding divisor smaller than √87 that we've already checked.

Therefore, using trial division, we've found that 87 is divisible by 1, 3, 29, and 87.

Method 2: Utilizing Divisibility Rules

Divisibility rules provide shortcuts for determining if a number is divisible by a specific integer without performing long division. Let's apply some key divisibility rules to 87:

• Divisibility Rule for 3: A number is divisible by 3 if the sum of its digits is divisible by 3. In the case of 87, 8 + 7 = 15, and 15 is divisible by 3 (15 ÷ 3 = 5). Therefore, 87 is divisible by 3.

• Divisibility Rule for 9: A number is divisible by 9 if the sum of its digits is divisible by 9. Since 15 (the sum of the digits of 87) is not divisible by 9, 87 is not divisible by 9.

• Divisibility Rule for 2: A number is divisible by 2 if its last digit is an even number (0, 2, 4, 6, or 8). Since the last digit of 87 is 7 (an odd number), 87 is not divisible by 2.

• Divisibility Rule for 5: A number is divisible by 5 if its last digit is 0 or 5. Since the last digit of 87 is 7, 87 is not divisible by 5.

• Divisibility Rule for 6: A number is divisible by 6 if it's divisible by both 2 and 3. Since 87 is not divisible by 2, it's not divisible by 6.

• Divisibility Rule for 10: A number is divisible by 10 if its last digit is 0. Since the last digit of 87 is 7, 87 is not divisible by 10.

Using these rules, we quickly confirm that 87 is divisible by 1 and 3. The remaining divisors (29 and 87) are found through the process of elimination or by recognizing that 29 is a prime number and a factor of 87.

Method 3: Prime Factorization

Prime factorization involves expressing a number as a product of its prime factors. This method helps identify all divisors systematically. The prime factorization of 87 is 3 x 29. Both 3 and 29 are prime numbers.

From the prime factorization, we can easily determine the divisors:

• 1 (because every number is divisible by 1)
• 3
• 29
• 87 (3 x 29)

Understanding the Divisors of 87

The divisors of 87 are 1, 3, 29, and 87. These numbers are all the integers that divide 87 without leaving a remainder. Notice that these divisors come in pairs: 1 and 87, and 3 and 29. This is because every divisor has a corresponding divisor such that their product is equal to the original number (87).

Frequently Asked Questions (FAQ)

• Q: Is 87 a prime number?

  • A: No, 87 is not a prime number. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Since 87 is divisible by 3 and 29, it's a composite number.

• Q: How can I find all the divisors of a larger number quickly?

  • A: For larger numbers, using prime factorization is the most efficient method. Finding the prime factors and then systematically combining them will give you all the divisors.

• Q: What is the significance of divisibility rules?

  • A: Divisibility rules save time and effort when determining whether a number is divisible by another. They are fundamental in simplifying calculations and understanding number properties.

• Q: Can a number have an odd number of divisors?

  • A: Yes, but only if the number is a perfect square. For example, the divisors of 16 (4 x 4) are 1, 2, 4, 8, and 16 (five divisors). A number that is not a perfect square will always have an even number of divisors.

• Q: Are there divisibility rules for all numbers?

  • A: While there are readily available and commonly used divisibility rules for many numbers, there isn't a universally established divisibility rule for every single number.

Conclusion: Mastering Divisibility

Determining what numbers 87 is divisible by provides a practical illustration of fundamental mathematical concepts. By employing methods like trial division, utilizing divisibility rules, and performing prime factorization, we effectively and efficiently found all the divisors of 87: 1, 3, 29, and 87. Understanding divisibility is crucial not only for solving basic arithmetic problems but also for tackling more advanced mathematical concepts and problem-solving scenarios. Remember that practice is key; the more you work with divisibility rules and different numerical methods, the stronger your understanding will become. So, keep exploring, keep practicing, and unlock the fascinating world of numbers!