d1, d2, and so on dn are divisors and the GCD of any two of them is 1;
This means that d1, d2 and so on are not multiples of each other.
All these numbers of the series themselves are factors of n and thus they divide n
22. If di, dr are all divisors of n and if gcd(di, dj) = 1 whenever...
Let (di, d2,... .dk) (w1, w2,..., wx)-0 (mod n) be an error detection scheme for the k-digit identification number did2 . . .dk, where 0 〈 di 〈 n. Prove that all transposition errors of two digits di and dj are detected if and only if gcd(ui-y,n) = 1 for i and j between 1 and k
Exercise 17.8 Find all positive integers n for which o(n) = 60. Whenever a mathematician asks you to find all numbers with certain properties, you are being asked to prove that the values you do find are the only values that work. Hint: Compare the factors of o(n) of the form 1+p+ ... + pk with the positive divisors of 60.
R R 5. To compute 1 = lim 2 COS dr and J = lim 22+1 sinc dx simultaneously .22 +1 R R0 R R using Residue Theorem, let f(x) 22 +1 C COSC sinc (1) Show that if z = x + iy, then Rf(R2) = and Sf(R2) = x2 +1 x2 +1 (2) Find Res[f, i]. (3) Show that I = 0 and J (4) Prove I = 0) in the above problem without using Residue Theorem. IT
T(n) is the number of divisors of n, and u(n)-1 Define an arithmetic function A as follows: if p is a prime and k 1 let A(p) log p for all other n, let A(n) 0. (Warning: A is NOT a multiplicative function!) Prove that (A* u)(n) log n for all n. (HINT: consider the various d which divide n expressed in terms of the prime factorization of n
HW: Show that the series __, an n=0 converges whenever ſal < 1, and diverges whenever al > 0.
Consider the following C codes to compute the gcd of two integers. /// code 1 #include <stdio.h> int gcd(int a, int b) { while (a != b) { if (a > b) a = a - b; else b = b - a; } return a; } /// code 2 #include <stdio.h> int getint() { int i; scanf("%d", &i); return i; } void putint(int i) { printf("%d\n", i); } int main()...
Let a and be be in . Show the following. If gcd(a,b)=1, then for every n in there exist x and y in such that n=ax+by. We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
3. Use Euclid's algorithm to compute the following. Show all your steps 1. gcd(781, 994) 2. gcd(67457, 43521)
QUESTION 4 Given the equation of a point, r(t) ( I)i ( -I)j Sketch the graph of r(r) = (1 + l)i + (r2-Dj fr-2 2. Draw the (a) t 4 marks) position vector r(0) on the same diagram. b) Find the unit tangent vector of the point at 0 and show it on the same diagram in (a). Explain what you understand about the direction of the tangent (5 marks)
Write the function get_factors(n) that returns a list containing all of the divisors (without remainder) of the integer n. Use the built-in function range() to create the appropriate sequence over which to run the list comprehension. Here are some usage examples: >>> get_factors(10) [1, 2, 5, 10] >>> get_factors(2017) # a prime number [1, 2017] >>> get_factors(60) # highly non-prime! why we have 60 minutes per hour [1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60] >>>...