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In Elementary Quantity Idea, it’s important to search out the biggest constructive integer that divides two or extra numbers with out the rest. For instance it’s helpful for lowering vulgar fractions to be in lowest phrases. To see an instance, to scale back 203/377 to lowest phrases, we have to know that 29 is the biggest constructive integer that divides 203 and 377. Then, we are able to write 203/377 = (7)(29)/(13)(29) = 7/13. How do we discover that 29 is the biggest integer that generally divides 203 and 377 ? A technique is by figuring out the prime factorizations of the 2 numbers and evaluating components. i.e. we have to know 203 = (7)(29) and 377 = (13)(29). A way more environment friendly technique is the Euclidean algorithm. The biggest constructive integer that divides two or extra numbers with out the rest known as the GREATEST COMMON FACTOR (G.C.F.) of the 2 or extra numbers. The primary technique of discovering G.C.F. is, by discovering the prime components of the numbers. The second technique based mostly on the Euclidean algorithm, is extra environment friendly and is mentioned right here. Its main significance is that it doesn’t require factoring. G.C.F. is also called Biggest Frequent Divisor, G.C.D. some occasions it’s also referred to as Highest Frequent Issue, H.C.F. I Technique based mostly on the Euclidean algorithm for locating G.C.F. of two numbers :
STEP 1 : Divide the larger quantity (Dividend) by the smaller quantity (Divisor) to get some The rest.
STEP 2 : Then divide the Divisor (turns into Dividend) by the The rest (turns into Divisor) to get a brand new The rest.
STEP 3 : Proceed the method of dividing the Divisors in succession by the Remainders acquired, until we get the The rest zero.
STEP 4 : The final Divisor is the G.C.F. of the given two numbers. All these steps are proven at one place as a single unit just like Lengthy Division. The tactic will probably be clear by the next examples.
Instance I(1) : Discover the G.C.F. of the numbers 16 and 30. Answer :
16 ) 30 ( 1
16
——
14 ) 16 ( 1
14 ——
G.C.F. 2 ) 14 ( 7
14
——-
0
——-
See the Biggest Frequent Issue discovering course of presentation given above.
STEP 1 : We divide the larger quantity (Dividend, 30) by the smaller quantity (Divisor, 16) to get The rest 14 (quotient being 1).
STEP 2 : Then, we divide the Divisor (16, turns into Dividend) by the The rest (14, turns into Divisor) to get a brand new The rest 2 (quotient being 1).
STEP 3 : We proceed the method of dividing the Divisors in succession by the Remainders acquired, until we get the The rest zero. we divide the Divisor (14, turns into Dividend) by the The rest (2, turns into Divisor) to get a brand new The rest 0 (quotient being 7). STEP 4 : The final Divisor, 2 is the G.C.F. of the given two numbers 16 and 30. Thus G.C.F. of 16 and 30 = 2. Ans.
Instance I(2) : Discover the G.C.F. of the numbers 45 and 120. Answer :
45 ) 120 ( 2
90
——
30 ) 45 ( 1
30
——
G.C.F. 15 ) 30 ( 2
30
——-
0
——-
See the G.C.F. discovering course of presentation given above. 120 is split by 45 to get 30 as the rest (quotient being 2). Within the subsequent stage, 30 is divisor and 45 is dividend. This division gave 15 as the rest (quotient being 1). Within the subsequent stage, 15 is divisor and 30 is dividend. This division gave 0 as the rest (quotient being 2). The final Divisor 15 is the G.C.F. of the given two numbers. Thus G.C.F. of 45 and 120 = 15. Ans.
Instance I(3) : Discover the G.C.F. of the numbers 1066 and 46189. Answer :
1066 ) 46189 ( 43
45838
——
351 ) 1066 ( 3
1053
——
G.C.F. 13 ) 351 ( 27
351
——-
0
——-
See the G.C.F. discovering course of presentation given above. 46189 is split by 1066 to get 351 as the rest (quotient being 43). Within the subsequent stage, 351 is divisor and 1066 is dividend. This division gave 13 as the rest (quotient being 3). Within the subsequent stage, 13 is divisor and 351 is dividend. This division gave 0 as the rest (quotient being 27). The final Divisor 13 is the G.C.F. of the given two numbers. Thus G.C.F. of 1066 and 46189 = 13. Ans. This division technique of discovering Biggest Frequent Issue is very helpful for locating the G.C.F.of enormous numbers. Think about doing this instance 3, by Prime Factorisation. You’ll realise the benefit of this division Course of over Prime Factorisation.
II Technique of discovering G.C.F. of greater than two numbers : As a way to discover the G.C.F. of greater than two numbers, first discover the G.C.F. of any two of them. Then, discover the G.C.F. of the third quantity and the G.C.F.of the primary two numbers, so obtained. Proceed this technique, so as, until all of the numbers are over. Allow us to see some Examples.
Instance II(1) : Discover the G.C.F. of the numbers 60, 90, 150. Answer : First, allow us to discover the G.C.F. of the numbers 60 and 90.
60 ) 90 ( 1
60
——
G.C.F. 30 ) 60 ( 2
60
——-
0
——-
Thus, G.C.F. of the numbers 60 and 90 = 30 Now allow us to discover the G.C.F. of 30 and 150. We will see 150 is 5 occasions 30. So, G.C.F. of 30 and 150 = 30. If one of many two numbers is an element of the opposite, then that issue is the G.C.F. of the 2 numbers. Thus, G.C.F. of the numbers 60, 90, 150 = 30. Ans.
Instance II(2) : Discover the G.C.F. of the numbers 70, 210, 315. Answer : First, allow us to discover the G.C.F. of the numbers 70 and 210. We will see 210 is 3 occasions 70. So, G.C.F. of 70 and 210 = 70. Now allow us to discover the G.C.F. of 70 and 315.
70 ) 315 ( 4
280
——
G.C.F. 35 ) 70 ( 2
70
——
0
——
Thus, G.C.F. of 70 and 315 = 35. So, G.C.F. of the numbers 70, 210, 315 = 35. Ans.
Instance II(3) : Discover the G.C.F. of the numbers 1197, 5320, 4389. Answer : First, allow us to discover the G.C.F. of the numbers 1197, 5320.
1197 ) 5320 ( 4
4788 ——
532 ) 1197 ( 2
1064
——
G.C.F. 133 ) 532 ( 4
532
——-
0
——-
Thus, G.C.F. of the numbers 1197 and 5320 = 133. Now allow us to discover the G.C.F. of 133 and 4389.
G.C.F. 133 ) 4389 ( 33
4389
——
0
——
Thus, the G.C.F. of 133 and 4389 = 133. So, The G.C.F. of the numbers 1197, 5320, 4389 = 133. Ans.
Instance II(4) : Discover the G.C.F. of the numbers 1701, 2106, 2754. Answer : First, allow us to discover the G.C.F. of the numbers 1701, 2106.
1701 ) 2106 ( 1
1701
——
405 ) 1701 ( 4
1620
——
G.C.F. 81 ) 405 ( 5
405
——-
0
——-
Thus, G.C.F. of the numbers 1701, 2106 = 81. Now allow us to discover the G.C.F. of 81 and 2754.
G.C.F. 81 ) 2754 ( 34
2754
——
0
——
Thus, the G.C.F. of 81 and 2754 = 81. SO, The G.C.F. of the numbers 1701, 2106, 2754 = 81. Ans.
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Source by Kalagara Venkata Lakshmi Narayana