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Introduction:

Definitions:

To better understand the concept of divisors, we need to first define a few related terms.

1. Integer: An integer is a whole number that can be positive, negative, or zero.
2. Divisor: A divisor of an integer n is a positive integer that divides n exactly, without leaving any remainder.
3. Factor: A factor of an integer n is any positive or negative integer that divides n exactly, without leaving any remainder. This means that a divisor is also a factor, but not all factors are divisors.
4. Multiple: A multiple of an integer n is any integer that can be obtained by multiplying n by another integer. For example, 2, 4, 6, 8, and so on are multiples of 2.
5. Proper divisor: A proper divisor of an integer n is a divisor of n that is not equal to n itself.

Examples:

Let us look at some examples to better understand the concept of divisors.

1. The divisors of 12 are 1, 2, 3, 4, 6, and 12.
2. The divisors of 17 are 1 and 17, as 17 is a prime number and only has two divisors.
3. The divisors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100.
4. The divisors of 0 are all integers, as any integer can divide 0 without leaving a remainder.
5. The divisors of -10 are 1, 2, 5, and 10, as well as -1, -2, -5, and -10.
6. The proper divisors of 28 are 1, 2, 4, 7, and 14.
7. The proper divisors of 13 are only 1, as 13 is a prime number.
8. The proper divisors of 100 are 1, 2, 4, 5, 10, 20, 25, and 50.
9. The multiples of 7 are 7, 14, 21, 28, and so on.
10. The multiples of -3 are -3, -6, -9, -12, and so on.

Quiz:

1. What is a divisor of an integer? A) A number that can be divided into another number without leaving a remainder. B) A positive integer that divides another integer exactly, without leaving a remainder. C) Any integer that divides another integer exactly, without leaving a remainder. D) A negative integer that divides another integer exactly, without leaving a remainder.
2. What is a factor of an integer? A) A positive integer that divides another integer exactly, without leaving a remainder. B) Any integer that can be divided into another integer without leaving a remainder. C) A negative integer that divides another integer exactly, without leaving a remainder. D) A positive or negative integer that divides another integer exactly, without leaving a remainder.
3. What is a proper divisor of an integer? A) A divisor of an integer that is not equal to the integer itself. B) A factor of an integer that is not equal to the integer-

4     What are the proper divisors of 18? A) 1, 2, 3, 6, and 18 B) 1, 2, 3, 4, 6, 8, 9, and 18 C) 1, 2, 3, 6, and 9 D) 1, 2, 3, 4, 6, 8, 12, and 18

5   What are the multiples of 5? A) 5, 10, 15, 20, 25, and so on B) 1, 5, 10, 15, 20, 25, and so on C) 2, 5, 8, 11, 14, 17, and so on D) 0, 5, 10, 15, 20, 25, and so on

6   What are the divisors of -15? A) 1, 3, 5, and 15 B) -1, -3, -5, and -15 C) 1, -1, 3, -3, 5, -5, 15, and -15 D) There are no divisors of a negative integer.

7   What are the proper divisors of 31? A) 1 B) 1 and 31 C) There are no proper divisors of a prime number. D) None of the above.

8   What are the factors of 42? A) 1, 2, 3, 6, 7, 14, 21, and 42 B) 1, 2, 3, 4, 5, 6, 7, 10, 14, 15, 21, 30, and 42 C) 1, 2, 3, 6, 7, 14, 21, 28, and 42 D) 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 28, 30, and 42

9  What are the multiples of -4? A) -4, -8, -12, -16, and so on B) -1, -4, -8, -12, -16, and so on C) 4, 8, 12, 16, and so on D) 0, -4, -8, -12, -16, and so on

FAQs:

• How do I find the divisors of an integer? To find the divisors of an integer, you can divide the integer by each of the positive integers less than or equal to the square root of the integer. If the result of the division is a whole number, then the divisor is a factor of the integer.

For example, to find the divisors of 36, we can divide it by 1, 2, 3, 4, 6, and 9. The divisors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.

• What is the difference between a divisor and a factor? A factor is a number that divides evenly into another number, whereas a divisor is a number by which another number is divided. In other words, a divisor is a factor of a number, but not all factors are divisors.

For example, 2 is a factor of 4, but it is also a divisor of 8. On the other hand, 3 is a factor of 12, but it is not a divisor of 7.

• Can negative numbers have divisors? Yes, negative numbers can have divisors just like positive numbers. The divisors of a negative number are also negative and positive integers that divide the negative number without leaving a remainder.

For example, the divisors of -12 are -1, -2, -3, -4, -6, and -12.

• What is the relationship between multiples and divisors? Multiples and divisors are related in that a multiple is the product of a divisor and another integer. For example, 15 is a multiple of 3 because 3 times 5 equals 15. Similarly, 3 and 5 are divisors of 15.
• Can an integer have an infinite number of divisors? No, an integer cannot have an infinite number of divisors. Every integer has a finite number of divisors, which is determined by its prime factorization.

For example, the number 12 has six divisors: 1, 2, 3, 4, 6, and 12. These are determined by the prime factorization of 12, which is 2^2 x 3.

Conclusion: