Download Understanding the Greatest Common Divisor (GCD) and Its Properties and more Study notes Mathematics in PDF only on Docsity!
!!!What is a the Greatest Common Divisor?
Before learning about the greatest common divisor, learn about the dividend, divisor and quotient of a number. If one number is divided by another number, then a result of the division is found. As a result, the number that is being divided is called the dividend. The divisor is the number that divides a given number. The number obtained as a result is known as the quotient. A divisor that does not completely divide a number produces a number known as the remainder.
The factor is a divisor that divides a number exactly, which means that the remainder will be zero. Every number has at least two factors, that is are, one and the number itself. Note that the factor is always less than or equal to the number which it divides. For example:,
- The factors of 4 are 1, 2, 4.
- The factors of 7 are 1, 7.
- The factors of 12 are 1, 2, 3, 4, 6, 12. The greatest common divisor (GCD) of two or more numbers is the most significant number among the factors of that those two or more numbers. And The greatest
common divisor of and is denoted by. The greatest common divisor is also the most significant common factor and highest common factor (HCF). If the greatest common divisor of two or more numbers is one, then the numbers are said to be co-prime. For example:,
- The greatest common divisors of 2 and 4 are is given as follows: The factors of 2 are 1, 2. The factors of 4 are 1, 2, 4. So, the common factors are 1, 2. The greatest greater number among 1 and, 2 is
Hence,. - The greatest common divisor of 5, and 13 is given as follows:. The factors of 5 are 1, 5. The factors of 13 are 1, 13. Here, the only common factor is 1. Hence,. So, 5 and 13 are co-prime.
- Consider the two rods of length 8 feet and 28 feet, respectively. Cut these rods into pieces of equal length. What will be the length of the longest piece that is common for both rods? Here, the length of the longest piece that is common for both the given rods is obtained by the GCD of the lengths of both rods. The factors of 8 are 1, 2, 4, 8. The factors of 28 are 1, 2, 4, 7, 14, 28. So, the common factors are 1, 2, and 4. Hence,. This process of finding GCD takes huge time for the largest numbers. We will discuss some other methods to find the GCD later. An integer is divisible by a non-zero integer if there is an integer such that and it is denoted by. Note that , here and the integer is. H3: Properties of the Greatest Common Divisor
For example,. Then, the linear combination as,is . Hence, here and. *For any positive integer ,. For example, And,and . Hence,. *If , and , then. For example, if and then, And,and
Hence,. *If , then. For example, if and then, . *If and , then. For example, if and then, , and 2 is an integer. Hence,. H3: Uses of the Greatest Common divisorDivisor Greatest The greatest common divisor of two or more numbers is very useful to in finding the least common multiple of those two numbers. The least common multiple of two or more numbers is a the least smallest number among the multiples of those two or more numbers. And The least common multiple of and is denoted by . For example, *Least The least common multiple of 12 and 40 is given as follows: Multiples The multiples of 12 are 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, …
Hence, from equations and , to we obtain, . In general, . Hence,. Note that if , then. Example : The LCM of 55 and 121 is given as follows: The numbers 55 and 121 can be written as, and. Here, the common factor is 11. That is,. Use,. Then, .
H2: The Prime Factorization Factorisation Method:
A number greater than 1 is called a prime or a prime number if its only factors are 1 and the number itself. Expressing a number into the product of its factors that are all prime numbers is called the prime factorization factorisation of a number. Note that by the fundamental theorem of arithmetic, any number greater thant 1 having has a unique prime factorizationfactorisation. The primes are 2, 3, 5, 7, 9, 11, 13, 17, 19, … *The prime factorization factorisation of a number 48 is given as follows: First, check whether the given number is divisible by the first prime 2 or not. Here, 48 is divisible by 2. That is,. Now, again check whether the result 24 is divisible by 2 or not. Here, 24 is divisible by 2. That is,. Similarly, check for 12., 12 is divisible by 2. That is,. And 6 is divisible by 2. That is,. Here, 3 is not divisible by 2. Hence, move to the next prime 3. And 3 is divisible by 3. That is,. Continue this process till we get 1 as the final result. Hence, the prime factorization factorisation of 48 is given by the factors of the above process. i.e.That is, (^). To fFind the GCD by using prime factorizationfactorisation Suppose tTo find the GCD of ,. First first, write the Prime prime factorization factorisation for. Then, find out the product of the common factors among them. Which This gives them. Example 1: Find the GCD of 56, 12, 90.
Then, is the last non-zero remainder in the division process, i.e.,. Example: Find the GCD of 42823 and 6409. Solution: Take the smallest value as. So, and. . Here, the last non-zero remainder is 17. Hence,.
H2: Example Problems
Example 1: Find the GCD of 357, 400 and 555. Solution: Write the prime factorization factorisation of 357, 400 and 555. That is,
There is no common factor. Hence,. Example 2: Simplify: . Solution: First, find the LCM of 8,6,4. The prime factorization factorisation of 8,6,4 is, . The common factor is 2. That is,. Use the formula, . Then,
Hence, the required answer is.
H2: Lesson Summary
*Relation The relation between the dividend, divisor and quotient is,
The greatest common divisor (GCD) of two or more numbers is a the greatest number among the factors of that those two or more numbers. And The greatest common divisor of and is denoted by.
The least common multiple (LCM) of two or more numbers is a the leassmallest number among the multiples of those two or more numbers. And The least common multiple of and is denoted by.
Expressing a number into the product of factors that are all prime numbers is called the prime factorization factorisation of a number. *Euclidean Algorithmalgorithm: For any the given integers and with , we make a continuous division algorithm, to obtain the system of equations.,
Then, is the last non-zero remainder in the division process, i.e.,.
