Divisors Of 144 A Comprehensive Guide To Finding Divisors

When diving into the world of mathematics, a fundamental concept to grasp is that of divisors. A divisor, quite simply, is a number that divides evenly into another number, leaving no remainder. To truly understand this concept, let's consider the question at hand: "Qual dos seguintes números é divisor de 144?" (Which of the following numbers is a divisor of 144?). The options given are A) 12, B) 15, C) 20, and D) 25. To answer this, we need to delve into the process of identifying divisors and then apply that knowledge to the number 144.

To begin, it's important to clarify what it means for a number to be a divisor. A divisor of a number is an integer that divides the number exactly, without leaving any remainder. For instance, 2 is a divisor of 6 because 6 ÷ 2 = 3, which is an integer. On the other hand, 4 is not a divisor of 6 because 6 ÷ 4 = 1.5, which is not an integer. With this understanding, we can now approach the task of identifying the divisors of 144. One method is to systematically test each number to see if it divides 144 without a remainder. This can be done through manual division or by using a calculator. However, to streamline the process, we can also look for patterns and rules that govern divisibility.

For example, we know that any even number is divisible by 2. Therefore, we can quickly check if 144 is divisible by 2, which it is, since 144 is an even number. Similarly, a number is divisible by 3 if the sum of its digits is divisible by 3. The sum of the digits of 144 is 1 + 4 + 4 = 9, which is divisible by 3, so 144 is also divisible by 3. These simple divisibility rules can help us narrow down the potential divisors more efficiently. Furthermore, understanding the prime factorization of a number can be incredibly useful in identifying its divisors. Prime factorization involves breaking down a number into its prime factors, which are the prime numbers that multiply together to give the original number. For 144, the prime factorization is 2^4 * 3^2, which means 144 = 2 * 2 * 2 * 2 * 3 * 3. Knowing the prime factors allows us to systematically construct all the divisors of 144 by combining different combinations of these prime factors. This approach is particularly helpful when dealing with larger numbers that have numerous divisors.

Now, let's evaluate the options provided: A) 12, B) 15, C) 20, and D) 25. We will determine which of these numbers is a divisor of 144 by applying the principles of divisibility and prime factorization discussed earlier.

Starting with option A, 12, we can ask ourselves: does 12 divide 144 evenly? One way to check this is to perform the division 144 ÷ 12. If the result is an integer, then 12 is indeed a divisor of 144. Performing the division, we find that 144 ÷ 12 = 12, which is an integer. Therefore, 12 is a divisor of 144. Alternatively, we can consider the prime factorization of 12, which is 2^2 * 3. Comparing this to the prime factorization of 144 (2^4 * 3^2), we see that 12's prime factors are included within 144's prime factors. This confirms that 12 is a divisor of 144.

Next, let's consider option B, 15. To check if 15 is a divisor of 144, we can divide 144 by 15. If the result is an integer, then 15 is a divisor. However, 144 ÷ 15 = 9.6, which is not an integer. Therefore, 15 is not a divisor of 144. Another way to approach this is to look at the prime factorization of 15, which is 3 * 5. While 144 has 3 as a prime factor, it does not have 5 as a prime factor. This means that 15 cannot divide 144 evenly.

Moving on to option C, 20, we can similarly check if 20 divides 144 evenly. Dividing 144 by 20 gives us 144 ÷ 20 = 7.2, which is not an integer. Thus, 20 is not a divisor of 144. Alternatively, the prime factorization of 20 is 2^2 * 5. Again, we see that while 144 has 2 as a prime factor, it does not have 5. This confirms that 20 is not a divisor of 144.

Finally, let's examine option D, 25. Dividing 144 by 25 gives us 144 ÷ 25 = 5.76, which is not an integer. Therefore, 25 is not a divisor of 144. The prime factorization of 25 is 5^2. Since 144 does not have 5 as a prime factor, it is clear that 25 cannot divide 144 evenly.

Identifying divisors of a number is a fundamental skill in mathematics, and there are several methods one can employ to accomplish this task. A systematic approach ensures that you don't miss any divisors and helps you understand the number's composition. Let's explore a step-by-step guide to identifying divisors, incorporating the use of prime factorization and divisibility rules.

Step 1: Start with Basic Divisors The first step in identifying divisors is to recognize that every number is divisible by 1 and itself. This is a basic rule of divisibility. For example, the number 144 is divisible by 1 and 144. These two divisors are always present, regardless of the number you're working with. Starting with these obvious divisors provides a foundation for finding the rest.

Step 2: Check Divisibility by 2 If the number is even, it is divisible by 2. This is one of the simplest divisibility rules and a quick way to identify a divisor. An even number is any number that ends in 0, 2, 4, 6, or 8. For instance, 144 is an even number because it ends in 4. Therefore, 2 is a divisor of 144. After identifying 2 as a divisor, you can divide the original number by 2 to find another factor. In this case, 144 ÷ 2 = 72, so 72 is also a divisor of 144.

Step 3: Apply the Divisibility Rule for 3 A number is divisible by 3 if the sum of its digits is divisible by 3. This rule is extremely useful for quickly determining whether 3 is a divisor. To apply this rule to 144, we add the digits: 1 + 4 + 4 = 9. Since 9 is divisible by 3 (9 ÷ 3 = 3), we know that 144 is also divisible by 3. To find the corresponding factor, we divide 144 by 3, which gives us 48. So, 3 and 48 are both divisors of 144.

Step 4: Check Divisibility by Other Prime Numbers After checking for divisibility by 2 and 3, it's useful to check for divisibility by other prime numbers such as 5, 7, 11, and so on. A number is divisible by 5 if its last digit is either 0 or 5. Since 144 ends in 4, it is not divisible by 5. There isn't a simple divisibility rule for 7, so you would typically perform the division to check. 144 ÷ 7 results in a quotient with a remainder, so 7 is not a divisor of 144. Similarly, you can check for divisibility by other prime numbers as needed.

Step 5: Use Prime Factorization Prime factorization is a powerful tool for identifying all the divisors of a number. Prime factorization involves expressing a number as a product of its prime factors. To find the prime factorization of 144, we can start by dividing it by its smallest prime factor, which is 2. 144 ÷ 2 = 72. Then, we divide 72 by 2 to get 36. We continue dividing by 2 until we can no longer do so evenly. 36 ÷ 2 = 18, 18 ÷ 2 = 9. Now, 9 is not divisible by 2, so we move to the next prime number, which is 3. 9 ÷ 3 = 3, and 3 ÷ 3 = 1. So, the prime factorization of 144 is 2^4 * 3^2. This means 144 = 2 * 2 * 2 * 2 * 3 * 3. Once you have the prime factorization, you can systematically list all the divisors by taking different combinations of the prime factors. For example, 2, 3, 22=4, 23=6, 222=8, 223=12, and so on.

Step 6: List All Divisors Systematically To ensure you've identified all divisors, it's helpful to list them systematically. Start with 1 and the number itself, then list the divisors you've found in pairs. For 144, we've identified the following divisors: 1 and 144, 2 and 72, 3 and 48. Continuing this process, we find the other divisors: 4 (144 ÷ 4 = 36), so 4 and 36 are divisors; 6 (144 ÷ 6 = 24), so 6 and 24 are divisors; 8 (144 ÷ 8 = 18), so 8 and 18 are divisors; 9 (144 ÷ 9 = 16), so 9 and 16 are divisors; and finally, 12 (144 ÷ 12 = 12), so 12 is a divisor. Listing these out, we have: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, and 144. These are all the divisors of 144.

After carefully analyzing the options and applying our knowledge of divisors, we can definitively answer the question: "Qual dos seguintes números é divisor de 144?" (Which of the following numbers is a divisor of 144?).

Based on our analysis, the correct answer is A) 12. We have shown that 144 ÷ 12 = 12, which is an integer, indicating that 12 divides 144 evenly. The other options, B) 15, C) 20, and D) 25, do not divide 144 without leaving a remainder. To further justify our answer, we considered the prime factorization of 144 (2^4 * 3^2) and confirmed that the prime factors of 12 (2^2 * 3) are contained within the prime factors of 144. This reinforces the conclusion that 12 is indeed a divisor of 144.

In summary, understanding the concept of divisors, applying divisibility rules, and utilizing prime factorization are crucial skills for identifying divisors of a number. By following these steps, we can confidently determine the divisors of any number and solve related problems with precision and accuracy.