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Definition of Divisor

A divisor is a number that divides another number exactly without leaving a remainder. For example, the number 4 is a divisor of 12 because 4 divides 12 exactly three times without leaving a remainder. Similarly, the number 5 is not a divisor of 12 because 5 does not divide 12 exactly.

The divisors of a number can be found by dividing the number by all the possible factors of the number. A factor is a number that can be multiplied by another number to get the original number. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. To find the divisors of 12, we divide 12 by each of its factors to see which ones divide 12 exactly without leaving a remainder.

Examples of Divisors

Example 1: Find the divisors of 24.

The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. To find the divisors of 24, we divide 24 by each of these factors to see which ones divide 24 exactly without leaving a remainder. The divisors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

Example 2: Determine whether 5 is a divisor of 30.

To determine whether 5 is a divisor of 30, we divide 30 by 5. If the division is exact, then 5 is a divisor of 30. If there is a remainder, then 5 is not a divisor of 30. 30 divided by 5 is 6, so 5 is a divisor of 30.

Example 3: Find the divisors of 48.

The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. To find the divisors of 48, we divide 48 by each of these factors to see which ones divide 48 exactly without leaving a remainder. The divisors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.

Example 4: Determine whether 7 is a divisor of 56.

To determine whether 7 is a divisor of 56, we divide 56 by 7. If the division is exact, then 7 is a divisor of 56. If there is a remainder, then 7 is not a divisor of 56. 56 divided by 7 is 8, so 7 is a divisor of 56.

Example 5: Find the divisors of 100.

The factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100. To find the divisors of 100, we divide 100 by each of these factors to see which ones divide 100 exactly without leaving a remainder. The divisors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100.

Example 6: Find the divisors of 24.
To find the divisors of 24, we list all the factors of 24, which are 1, 2, 3, 4, 6, 8, 12, and 24. Therefore, the divisors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

Example 7: Find the divisors of 37.
To find the divisors of 37, we note that 37 is a prime number, and the only divisors of a prime number are 1 and the number itself. Therefore, the divisors of 37 are 1 and 37.

Example 8: Find the divisors of 100.
To find the divisors of 100, we list all the factors of 100, which are 1, 2, 4, 5, 10, 20, 25, 50, and 100. Therefore, the divisors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100.

Example 9: Find the divisors of 144.
To find the divisors of 144, we list all the factors of 144, which are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, and 144. Therefore, the divisors of 144 are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, and 144.

Example 10: Find the divisors of 17.
To find the divisors of 17, we note that 17 is a prime number, and the only divisors of a prime number are 1 and the number itself. Therefore, the divisors of 17 are 1 and 17.

These examples demonstrate that finding divisors is relatively simple once we identify the factors of a given number. Divisors can provide us with important information about a number, such as its prime factorization, properties, and applications in problem-solving.

Quiz on Divisors

Now that we’ve covered the basics of divisors, it’s time to test your knowledge with a quiz. Below are 10 questions to see how well you understand the concept of divisors.

1. What is a divisor? a. A number that is divided b. A number that divides another number exactly without leaving a remainder c. A number that is subtracted
2. What are the divisors of 24? a. 1, 2, 3, 4, 6, 8, 12, and 24 b. 1, 3, 5, 8, 15, and 24 c. 2, 4, 6, 8, 10, 12, 16, and 20
3. Is 7 a divisor of 28? a. Yes b. No
4. What are the divisors of 36? a. 1, 2, 3, 4, 6, 9, 12, 18, and 36 b. 1, 2, 4, 9, 12, and 18 c. 2, 3, 6, 9, 12, and 18
5. Is 8 a divisor of 56? a. Yes b. No
6. What are the divisors of 60? a. 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60 b. 1, 3, 5, 12, 15, and 30 c. 2, 4, 6, 10, 12, 20, and 30
7. Is 9 a divisor of 45? a. Yes b. No
8. What are the divisors of 72? a. 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72 b. 1, 2, 4, 6, 8, 12, 18, 24, and 36 c. 2, 3, 4, 6, 8, 9, 12, 18, and 24
9. Is 11 a divisor of 99? a. Yes b. No
10. What are the divisors of 90? a. 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90 b. 1, 3, 5, 9, 10, and 15 c. 2, 4, 5, 10, 15, and 30

Answers:

1. b
2. a
3. a
4. a
5. a
6. a
7. a
8. a
9. b
10. a

Conclusion

FAQs

Q: What is the difference between a divisor and a factor? A: In mathematics, a divisor is a number that divides evenly into another number, while a factor is a number that is multiplied by another number to get a product. Divisors are related to factors because if a number is divisible by another number, then that number is also a factor of the first number.

Q: How do you find the divisors of a number? A: To find the divisors of a number, you can list all the numbers that divide evenly into that number. For example, to find the divisors of 12, you would list 1, 2, 3, 4, 6, and 12.

Q: What is the purpose of finding divisors? A: Divisors are useful in many areas of mathematics, including number theory, algebra, and calculus. They are used to find factors of a number, determine whether a number is prime or composite, and to simplify fractions.

Q: What is the relationship between divisors and multiples? A: A multiple of a number is a number that is the product of that number and another integer. For example, the multiples of 3 are 3, 6, 9, 12, and so on. Divisors and multiples are related because if a number is a multiple of another number, then that number is also divisible by the first number. For example, 12 is a multiple of 3, so 12 is also divisible by 3.