Give an example of two distinct positive integers m, n such that n^2 is a multiple of m, but n is not a multiple of m. How about having m be less than n?
n = 8,
m = 16
n is not multiple of m, But n^2 = 64 is multiple of m. Hence 16, 8
is the answer for this..
Give an example of two distinct positive integers m, n such that n^2 is a multiple...
Given an unsorted array of distinct positive integers A [ 1......n ] in the range between 1 and 10000 and an integer i in the sane range. Here n can be arbitrary large You want to find out whether there are 2 elements of the array that add up to i. Give an algorithm that runs in time (O(n).
(Erdos, distinct sums problem). k positive integers x_1,...,x_k are said to have distinct sums if the sum of any subset of them are different. Let f(n) denote the maximal k for which there exists such set within [n]. (a) Come up with a simple example of a set in [n] of size log_2 n. (Hint: binary expansion..) (b) Show that n >= (2^k * 2/3 - 1) / sqrt(3k)
8. Define (n) to be the number of positive integers less than n and n. That is, (n) = {x e Z; 1 < x< n and gcd(x, n) = 1}|. Notice that U (n) |= ¢(n). For example U( 10) = {1, 3,7, 9} and therefore (10)= 4. It is well known that (n) is multiplicative. That is, if m, n are (mn) (m)¢(n). In general, (p") p" -p Also it's well known that there are relatively prime, then...
Prove that for all integers n, (-n) mod 2 = n mod 2. Give an example to show that it is not always true that (-n) mod 3 = n mod 3. Professor mentioned to prove for odd and even integers, however, I don't know how to start the proof.
(3.5) Summing the Euler S-function (n): The Euler 6-function 6(n) counts the number of positive integers less than or equal to n, which are relatively prime with n. Evaluate 4(d), and prove that your answer is correct. (3.4) Relatively Prime Numbers and the Chinese Re- mainder Theorem: Give an example of three positive integers m, n, and r, and three integers a, b, and c such that the GCD of m, n, and r is 1, but there is no...
Find the smallest positive integer that has precisely n distinct prime divisors. 'Distinct prime divisor'Example: the prime factorization of 8 is 2 * 2 * 2, so it has one distinct prime divisor. Another: the prime factorization of 12 is 2 * 2 * 3, so it has two distinct prime divisors. A third: 30 = 2 * 3 * 5, which gives it three distinct prime divisors. (n = 24 ⇒ 23768741896345550770650537601358310. From this you conclude that you cannot...
number thoery
just need 2 answered
2. Let n be a positive integer. Denote the number of positive integers less than n and rela- tively prime to n by p(n). Let a, b be positive integers such that ged(a,n) god(b,n)-1 Consider the set s, = {(a), (ba), (ba), ) (see Prollern 1). Let s-A]. Show that slp(n). 1. Let a, b, c, and n be positive integers such that gcd(a, n) = gcd(b, n) = gcd(c, n) = 1 If...
Let A = {2, 3, . . . , 50}, that is, A is the set of positive
integers greater than 1 and less than 51. Determine the smallest
number x such that every subset of A having x elements contains at
least two integers that have a common divisor greater than 1, and
justify your answer.
(5 marks) Let A {2,3, ,50}, that is, A is the set of positive integers greater than 1 and less than 51. Determine...
Prove: If n and a are positive integers and n=(a^2+ 1)/2, then n is the sum of the squares of two consecutive integers (that is, n=k^2+ (k+1)^2 for some integer k).
PLEASE SHOW ALL STEPS WITH EXPLAINATION Let m and n be positive integers and let k be the least common multiple of m and n. Show that mZ∩nZ=kZ.