What Is Divisible By 57

What is Divisible by 57? Unlocking the Secrets of Divisibility

Divisibility rules are fundamental concepts in mathematics, providing shortcuts for determining whether a number is evenly divisible by another without performing long division. While many are familiar with the rules for 2, 3, 5, and 10, others, like divisibility by 57, might seem less intuitive. This comprehensive guide will explore the intricacies of divisibility by 57, offering practical methods, illustrative examples, and a deeper understanding of the underlying mathematical principles. We'll delve into why some numbers are divisible by 57 and others aren't, examining the factors of 57 and their implications for divisibility tests.

Understanding Divisibility and Factors

Before diving into the specifics of divisibility by 57, let's establish a solid foundation. A number is divisible by another if the result of their division is a whole number, leaving no remainder. For example, 12 is divisible by 3 because 12 ÷ 3 = 4. The numbers that divide evenly into a given number are called its factors or divisors. For instance, the factors of 12 are 1, 2, 3, 4, 6, and 12.

Finding the factors of a number is crucial to understanding divisibility. The prime factorization of a number expresses it as the product of its prime factors (numbers divisible only by 1 and themselves). For example, the prime factorization of 12 is 2 x 2 x 3 (or 2² x 3). This prime factorization helps us understand which numbers will divide evenly into 12.

Prime Factorization of 57: The Key to Divisibility

The first step in understanding divisibility by 57 is to find its prime factorization. 57 can be expressed as 3 x 19. Both 3 and 19 are prime numbers. This prime factorization reveals the key to determining divisibility by 57: a number is divisible by 57 if and only if it is divisible by both 3 and 19. This is because divisibility is transitive; if a number is divisible by a and b, and a and b are coprime (share no common factors other than 1), then it's divisible by their product (a x b). Since 3 and 19 are coprime, a number divisible by both is necessarily divisible by 57.

Practical Methods for Checking Divisibility by 57

Now that we understand the underlying principle, let's explore practical methods for determining whether a number is divisible by 57. We can't rely on a single, simple rule like we can for some other numbers. Instead, we need a two-step process:

Step 1: Check for Divisibility by 3

The divisibility rule for 3 is well-known: a number is divisible by 3 if the sum of its digits is divisible by 3. For example, let's consider the number 171. The sum of its digits is 1 + 7 + 1 = 9, which is divisible by 3. Therefore, 171 is divisible by 3.

Step 2: Check for Divisibility by 19

Divisibility by 19 is less straightforward. There isn't a simple digit-sum rule. However, we can use a more algorithmic approach. Let's illustrate with the number 171 (which we've already established is divisible by 3):

• Method 1: Long Division: The most reliable method, though potentially tedious for larger numbers, is simply to perform long division. If the division results in a whole number with no remainder, the number is divisible by 19. 171 ÷ 19 = 9. Therefore, 171 is divisible by 19.

• Method 2: Subtraction Method (for smaller numbers): For smaller numbers, you can use a subtraction method. Repeatedly subtract multiples of 19 until you reach 0 or a number clearly divisible or indivisible by 19. Let’s try it with 171:

  • 171 - 19 = 152
  • 152 - 19 = 133
  • 133 - 19 = 114
  • 114 - 19 = 95
  • 95 - 19 = 76
  • 76 - 19 = 57
  • 57 - 19 = 38
  • 38 - 19 = 19
  • 19 - 19 = 0

Since we reached 0, 171 is divisible by 19. This method can become cumbersome for larger numbers.

Conclusion of the Two-Step Process: Since 171 is divisible by both 3 and 19, it is divisible by 57 (57 x 3 = 171).

Illustrative Examples: Numbers Divisible by 57

Let's examine a few more examples to solidify our understanding:

• 342: The sum of the digits (3 + 4 + 2 = 9) is divisible by 3. 342 ÷ 19 = 18. Therefore, 342 is divisible by 57 (57 x 6 = 342).

• 114: The sum of digits (1+1+4 = 6) is divisible by 3. 114 ÷ 19 = 6. Therefore, 114 is divisible by 57 (57 x 2 = 114).

• 663: The sum of digits (6+6+3 = 15) is divisible by 3. 663 ÷ 19 = 35. Therefore 663 is divisible by 57 (57 x 11 = 627 - it is not divisible by 57, shows there's possibility of mistake in calculations)

• 912: The sum of digits (9 + 1 + 2 = 12) is divisible by 3. 912 ÷ 19 = 48. Therefore 912 is divisible by 57 (57 x 16 = 912)

• 1449: The sum of digits (1 + 4 + 4 + 9 = 18) is divisible by 3. 1449 ÷ 19 = 76 + 5/19. This means that although it's divisible by 3, it's not divisible by 19 and therefore not divisible by 57.

Dealing with Larger Numbers

For larger numbers, the long division method for checking divisibility by 19 becomes the most practical. While there are more advanced divisibility rules for 19 involving modular arithmetic, they are generally more complex than simply performing the division. Modern calculators make this process significantly easier.

Frequently Asked Questions (FAQ)

Q1: Is there a single, simple divisibility rule for 57?

A1: No, there isn't a single, simple rule like the sum-of-digits rule for 3. The process involves checking divisibility by both 3 and 19.

Q2: Why is checking for divisibility by both 3 and 19 necessary?

A2: Because 57 is the product of the prime numbers 3 and 19. A number must be divisible by both of these prime factors to be divisible by 57.

Q3: What if a number is divisible by 3 but not by 19?

A3: Then the number is not divisible by 57. Both conditions must be met.

Q4: Are there any shortcuts for checking divisibility by 19 besides long division?

A4: While there are more advanced methods using modular arithmetic, they often add complexity and are not significantly faster than long division for most practical purposes, especially without access to specialized software.

Conclusion: Mastering Divisibility by 57

Understanding divisibility by 57, while initially seeming challenging, boils down to a systematic two-step process: verifying divisibility by 3 (using the sum-of-digits rule) and then verifying divisibility by 19 (most efficiently using long division). This approach, rooted in the prime factorization of 57, provides a robust and reliable method for determining divisibility, applicable to numbers of all sizes. By breaking down the problem into smaller, manageable parts and leveraging familiar divisibility rules, we can confidently tackle the seemingly complex task of identifying numbers divisible by 57. This understanding not only enhances mathematical skills but also illustrates the power of prime factorization in unlocking the secrets of divisibility. Remember, practice is key to mastering these techniques. The more you work with these methods, the quicker and more intuitive they will become.