Multiple – a multiple of a number is a number that can be gotten from the product of the number and any integer.

Example 1: In the case of the number 5, multiples of 5 would include but are not limited to: -5, 0, 5, 10, 15, …

Example 2: In the case of the number 37, multiples of 5 would include but are not limited to: -37, 0, 37, 74, …

Divisor – a divisor is a factor of a given number that can evenly divide said number. To evenly divide a number would to leave a quotient without any remainders.

Example 1: In the case of the number 6, the divisors of 6 would be any of its factors, 1, 2, 3, or 6, where the quotients would be 6, 3, 2, and 1 respectively.

Example 2: In the case of the number 7, the divisors of 7 would be any of its factors of 1 or 7, with the respective quotients being 7 and 1.

Side note: As divisors include all of the number’s factors, all the negative counterparts of the factors are also divisors.

Difference: So, what is the difference between a multiple and a divisor? Firstly, a multiple of a number is the product of the given number and any integer, making multiples numbers that can potentially be greater than the original number. On the other hand, Divisors are numbers that divide the given number evenly. Therefore, Divisors are numbers limited to the factors of the given number. One major difference that can be observed are the magnitude of the two values, where multiples tend to have a specific, but infinite domain, whereas divisors follow a finite and also specific set of rules.

GCD (Greatest Common Denominator) – The GCD between two numbers is the greatest factor that is shared between two numbers.

Example 1: 18 & 28

Factors of 18: 1, 2, 3, 6, 18

Factors of 28: 1, 2 ,4, 7, 14, 28

In the case of 18 and 28, we can determine that the greatest value shared by 18 and 28 within its factors is 2. Therefore, the GCD of 18 and 28 is 2.

Example 2: 27 & 54

Factors of 27: 1, 3, 9, 27

Factors of 54: 1, 2 , 3, 6, 9, 18, 27, 54

In the case of 27 and 54, we can determine that the greatest value shared by 27 and 54 within its factors is 27. Therefore, the GCD of 27 and 54 is 27.

LCM (Least Common Multiple) – The LCM between two numbers is the smallest positive number shared by the multiples of two numbers.

Example 1: 1 & 8

Multiples of 1: 1, 2, 3, 4, 5, 6, 7, 8, 9, …

Multiples of 8: 8, 16, …

In the case of 1 and 8, the smallest positive multiple shared by 1 and 8 is the number 8. Therefore, the LCM of 1 and 8 is 8.

Example 2: 24 & 39

Multiples of 24: 24, 48, 72, 96, 120, 144, 168, 192, 216, 240, 264, 288, 312, 336, …

Multiples of 39: 39, 78, 117, 156, 195, 234, 273, 312, 351, …

In the case of 24 and 39, the smallest positive multiple shared by 24 and 39 is the number 312. Therefore, the LCM of 24 and 39 is 312.

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