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Greatest Common Factor – Finding Made Easy – Lucid Explanation of an Efficient Method With Examples
In elementary number theory, it is important to find the largest positive integer that divides two or more numbers with no remainder. For example, it is useful to reduce vulgar fractions to be in lowest terms. To see an example, to reduce 203/377 to the lowest terms, we need to know that 29 is the largest positive integer that divides 203 and 377. Then we can write 203/377 = (7)(29)/( 13) (29) = 7/13. How do you find that 29 is the largest integer that commonly divides 203 and 377? One way is to determine the prime factorizations of the two numbers and compare the factors. that is, we need to know 203 = (7)(29) and 377 = (13)(29). A much more efficient method is Euclid’s algorithm. The largest positive integer that divides two or more numbers with no remainder is called the GREATEST COMMON FACTOR (GCF) of the two or more numbers. The first method to find GCF is to find the prime factors of numbers. The second method, based on Euclid’s algorithm, is more efficient and is discussed here. Its major advantage is that it does not require factoring. GCF is also known as Greatest Common Divisor, GCD sometimes it is also called Highest Common Factor, HCF I Euclidean algorithm based method to find GCF of two numbers:
STEP 1: Divide the larger number (dividend) by the smaller number (divisor) to get a remainder.
STEP 2: Then divide the Divisor (becomes Dividend) by the Remainder (becomes Divisor) to get a new Remainder.
STEP 3: Continue the process of successively dividing the divisors by the obtained remainders, until we get zero remainder.
STEP 4: The last divisor is the PGCF of the two given numbers. All of these steps are displayed in one place as a single unit similar to the long division. The method will be clear by the following examples.
Example I(1): Find the PGCF of the numbers 16 and 30. Solution:
16 ) 30 ( 1
16
——
14 ) 16 ( 1
14 ——
CPF 2 ) 14 ( 7
14
——-
0
——-
See the presentation of the process of finding the greatest common factor given above.
STEP 1: We divide the larger number (Dividend, 30) by the smaller number (Divisor, 16) to get the Remainder 14 (the quotient being 1).
STEP 2: Then, we divide the Divisor (16, becomes Dividend) by the Remainder (14, becomes Divisor) to obtain a new Remainder 2 (the quotient being 1).
STEP 3: We continue the process of dividing the divisors successively by the remainders obtained, until we obtain the remainder zero. we divide the Divisor (14, becomes Dividend) by the Remainder (2, becomes Divisor) to obtain a new Remainder 0 (the quotient being 7). STEP 4: The last divisor, 2 is the PGCF of the two given numbers 16 and 30. So PGCF of 16 and 30 = 2. Rep.
Example I(2): Find the PGCF of the numbers 45 and 120. Solution:
45) 120 ( 2
90
——
30 ) 45 ( 1
30
——
BCF 15 ) 30 ( 2
30
——-
0
——-
See the overview of the GCF research process given above. 120 is divided by 45 to get 30 as the remainder (the quotient being 2). In the next step, 30 is the divisor and 45 is the dividend. This division gave 15 as the remainder (the quotient being 1). In the next step, 15 is the divisor and 30 is the dividend. This division gave 0 as the remainder (the quotient being 2). The last divisor 15 is the PGCF of the two given numbers. Thus GCF of 45 and 120 = 15. Rep.
Example I(3): Find the PGCF of the numbers 1066 and 46189. Solution:
1066) 46189 ( 43
45838
——
351) 1066 ( 3
1053
——
BCF 13 ) 351 ( 27
351
——-
0
——-
See the overview of the GCF research process given above. 46189 is divided by 1066 to obtain 351 as the remainder (the quotient being 43). In the next step, 351 is the divisor and 1066 is the dividend. This division gave 13 as the remainder (the quotient being 3). In the next step, 13 is the divisor and 351 is the dividend. This division gave 0 as remainder (quotient being 27). The last divisor 13 is the PGCF of the two given numbers. Thus GCF of 1066 and 46189 = 13. Rep. This method of dividing to find the greatest common factor is particularly useful for finding the PGCF of large numbers. Imagine doing this example 3, by Prime Factorization. You will realize the advantage of this division process over prime factorization.
II Method to find the PGCF of more than two numbers: In order to find the PGCF of more than two numbers, first find the PGCF of two of them. Next, find the PGCF of the third number and the PGCF of the first two numbers, thus obtained. Continue this method, in order, until all numbers are completed. Let’s see some examples.
Example II(1): Find the PGCF of the numbers 60, 90, 150. Solution: First find the PGCF of the numbers 60 and 90.
60 ) 90 ( 1
60
——
BCF 30 ) 60 ( 2
60
——-
0
——-
So PGCF of numbers 60 and 90 = 30 Now find the PGCF of 30 and 150. We can see that 150 is 5 times 30. So PGCF of 30 and 150 = 30. If either of the two numbers is a factor of the other, then that factor is the PGCF of the two numbers. Thus, GCF of the numbers 60, 90, 150 = 30. Rep.
Example II(2): Find the PGCF of the numbers 70, 210, 315. Solution: First, find the PGCF of the numbers 70 and 210. We can see that 210 is 3 times 70. So, PGCF of 70 and 210 = 70. Now find the GCF of 70 and 315.
70) 315 (4
280
——
BCF 35 ) 70 ( 2
70
——
0
——
So GCF of 70 and 315 = 35. So GCF of numbers 70, 210, 315 = 35. Rep.
Example II(3): Find the PGCF of the numbers 1197, 5320, 4389. Solution: First find the PGCF of the numbers 1197, 5320.
1197) 5320 ( 4
4788 ——
532 ) 1197 ( 2
1064
——
BCF 133 ) 532 ( 4
532
——-
0
——-
So, GCF of the numbers 1197 and 5320 = 133. Now find the GCF of 133 and 4389.
FBC 133 ) 4389 ( 33
4389
——
0
——
So the GCF of 133 and 4389 = 133. So the GCF of the numbers 1197, 5320, 4389 = 133. Rep.
Example II(4): Find the PGCF of the numbers 1701, 2106, 2754. Solution: First find the PGCF of the numbers 1701, 2106.
1701 ) 2106 ( 1
1701
——
405) 1701 (4
1620
——
BCF 81 ) 405 ( 5
405
——-
0
——-
So, GCF of the numbers 1701, 2106 = 81. Now find the GCF of 81 and 2754.
FBC 81 ) 2754 ( 34
2754
——
0
——
Thus, the GCF of 81 and 2754 = 81. SO, The GCF of the numbers 1701, 2106, 2754 = 81. Rep.
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