Euler's Theorenm and Remainder problems.
Hello! I am Harsh Sharma B.Tech in Computer Science CAT 2016-98 percentile (OA) Aptitude Trainer/Tutor = Movies | Music | Football 2
1. Euler Number Let's start with first understanding the basic concepts of Euler Number.
4L The Euler number of a natural number x means the number of natural numbers which are less than x and are co-prime to x. E.g. The Euler number of 6 will be 2 as the natural numbers 1 & 5 are the only two numbers which are less than 6 and are also co-prime to 6. 4
Example Find Euler Number of 15? Solution - The natural numbers which are less than 15 and are also co-prime to 15 are:- 1,2,4,7,8,11,13,14 So, Euler Number f 15 = 8 5
Formula Mathematically, the Euler number of a number Z denoted by the symbol E(Z) is calculated as explained below E(Z) = Z(1-1/P) (1-1/Q) (1-1/R) where P, Q and R are the different prime factors of Z.
Question- Find Euler Number of 72? 21 2 236 218 E(+z) = 72( 1-2)/ 1-3 7
Example Find Euler Number of 15? Solution-> 15=3*5 So, E(15) = 15(1-1/3) (1-15) -15(2/3) (4/5)
Example Find Euler Number of 19? Solution-> 19=1+19 So, E(19) = 19(1-1/19) - 19(18/19) = 18
Since for a number p' to be prime, the necessary condition is that it should not be divisible by any number other then 1 and itself. So, As a general rule of thumb we can state that Euler number of a prime number will involve all the natural numbers less then it. Hence, Euler Number(prime:p')=p-1 Example->E(13)=12, E(17-16, E(19)=18 and so on. 10
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