When write complete framework "Decreases method", there is a details to note: The results from the function to return, to the result assignment in the main function to a variable in order to follow the code used, or even call that function, but also not the result of the function execution. Because the result returned is a local variable, the function can not cross-use. In that moment to run a custom function in a timely manner will result assigned to a variable
Preface:
And multiple divisors: There exist in the relationship between two integers divisible. That is, if a divisible b, i.e. a% b = 0 (remainder), then a is a multiple of b, b is a number from about a (large is a multiple, small of about several multiples greater than the number equal to about). Such as: 12 ÷ 3 = 4 ... 0, 12 is a multiple of 3, the number of about 12 3
Divisor: Number of public about a few integers
The greatest common divisor
: Greatest Common Divisor (GCD) refers to two or more integer numbers of about a maximum. a, b is referred to as the greatest common divisor (a, b), the same, a, b, c referred to as the greatest common divisor (a, b, c), the greatest common divisor of integers have the same sign. Greatest common divisor There are several ways, a common prime factor decomposition method, the short division, Euclidean, Decreases method. And the greatest common divisor corresponding to the least common multiple concepts, a, b referred to as the least common multiple of [a, b]
Examples of 1:12 and 16
There are about 12 ∵ number 1,2,3,4,6,12, and 16 is about several 1,2,4,8,16. (An integer divisors are limited)
∴ divisor: 1,2,4
∴ greatest common divisor: 4
Referred to as (12, 16) = 4, or as (16,12) = 4
The least common multiple
: Least Common Multiple (LCM) two or more integer multiples of the public is called a common multiple thereof, wherein other than 0 smallest common multiple of a least common multiple of these is called an integer. Integer a, b referred to as the least common multiple of [a, b], the same, a, b, c is referred to as the least common multiple [a, b, c], the plurality of the least common multiple of an integer of the same sign has
Examples of 2:12 and 16
There 12,24,36,48,60,72,84,96,108 ...... in multiples of 12, and a multiple of 16 have 16,32,48,64,80,96,112 .... .., (an integer multiple of the infinite number
∴ have a common multiple of 48, 96, ......
∴ least common multiple: 48
Referred to as [12, 16] = 48 or [16, 12] = 48
Method for finding the greatest common divisor:
(1) prime factor decomposition method (all are all prime factors)
(2) :( Euclidean Euclidean algorithm)
Programming ideas: a ÷ b = c ... d (where a> b), when d is 0, then b is the greatest common divisor would otherwise be removed with d b, b case the dividend, divisor d, and in the process, from start to finish, always a symbol representative of the dividend, the divisor b representatives, representative of C supplier, d represents the number of I, b so as to dividend, will become a, i.e. a = b, and d to when the divisor, It becomes necessary to b, i.e., b = d
. 1 DEF GCD (A, B): # define the greatest common divisor function 2 . 3 IF A < B: . 4 . 5 A, B = B, A . 6 . 7 R & lt. 1 = # as long as the initial value is not 0 can be . 8 . 9 the while = R & lt! 0: 10 . 11 R & lt% = a B 12 is 13 is a = B 14 15 B = R & lt 16 . 17 return a 18 is . 19 20 is 21 is DEF LMG (a, B): # define the least common multiple of the demand function 22 is 23 is return A // B * GCD (A, B) 24 25 26 is 27 DEF main (): 28 29 A = the eval (INPUT ()) 30 31 is B = the eval (INPUT ()) 32 33 is Print ( " Maximum Convention number: " , GCD (A, B)) 34 is 35 Print ( " least common multiple: " , LCM (A, B)) 36 37 [ main ()
The code can also function without the main line (i.e., remove DEF main () and main () these two), after removing not only the implementation of a custom function. But in order to complete rigorous, clear thinking, it is recommended into the main function
(3) Method Decreases:
Notes: Euclidean greatest common divisor efficient
The least common multiple of the Dharma:
Method a: See above (1)
Method two: See above (2), in addition to seeking the greatest common divisor by (a, b) × [a, b] = a × b greatest common divisor (Formula Act)
Notes: Formula to find the least common multiple efficient
. 1 a = the eval (INPUT ()) 2 . 3 B = the eval (INPUT ()) . 4 . 5 IF a < B: . 6 . 7 a, B = B, a . 8 . 9 # ensure a is not less than B 10 . 11 12 is COUNT = 0 13 is 14 IF a% 2 == 0 and b% 2 == 0: # a and b all even, the following code is executed 15 16 the while a% 2 == 0 and b% 2 == 0: # If all even , removing constantly with a 2 and B, until the failure can be an even number (but not necessarily coprime, such as 11 and 77) . 17 18 is a = int (a / 2) #Division returns a float type, need turned int . 19 20 is B = int (B / 2 ) 21 is 22 is COUNT + =. 1 # statistics except 2 how many times 23 is 24 the while ab & = B:! # Important: throughout the program, a is always expressed minuend, b is the subtrahend, ab is poor. And must ensure that no less than a B, the following statement would if this feature is implemented 25 26 is a, B = B, ab & # minuend is calculated for each cycle and a difference between the minuend and subtrahend into assignments, so that the next calculation 27 28 IF a < B: 29 30 a, B = B, a 31 is 32 Print (B * 2 ** count) # multiplied by the power of the finally obtained count subtrahend and 2 33 is 34 is 35 the else : # A and b whether all even, perform the following at 36 37 [ the while ab & =! B: 38 is 39 A, b = b, ab & 40 41 is IF A < b: 42 is 43 is A, b = b, A 44 is 45 Print (b)
Full frame (function package):
1 def abc(a,b): 2 if a<b: 3 a,b = b,a 4 while a-b != b: 5 a,b = b,a-b 6 if a<b: 7 a,b = b,a 8 return b 9 10 def main(): 11 a = eval(input()) 12 b = eval(input()) 13 count = 0 14 if a%2 == 0 and b%2 ==0: 15 the while A% 2 == 0 and B% 2 == 0: 16 A = int (A / 2 ) . 17 B = int (B / 2 ) 18 is COUNT + =. 1 . 19 B = ABC (A, B) 20 is Print ( " The greatest common divisor: " , B * 2 ** COUNT) 21 is the else : 22 is B = ABC (A, B) 23 is Print ( " the greatest common divisor: " , B) 24 main ()
Important: In the main function, the result of the function ABC () to be assigned to a variable, or even return b, b the subsequent code does not take too, but with the original b