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Q18 12 Points For any positive integer n, let bn denote the number of n-digit positive...
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...
Discrete Mathematics 7. (15 points) Let an be the number of length n ({ne Zin 20}) ternary strings (strings made up of {0, 1, 2), ex. 01211120002) that contain two consecutive digits that are the same. For example, a = 3 since the only ternary strings of length 2 with matching consecutive digits are 00, 11, and 22. Also, a, = 0, since in order to have consecutive matching digits, a string must be of length at least two. a....
I got a C++ problem. Let n be a positive integer and let S(n) denote the number of divisors of n. For example, S(1)- 1, S(4)-3, S(6)-4 A positive integer p is called antiprime if S(n)くS(p) for all positive n 〈P. In other words, an antiprime is a number that has a larger number of divisors than any number smaller than itself. Given a positive integer b, your program should output the largest antiprime that is less than or equal...
1. (Integers: primes, divisibility, parity.) (a) Let n be a positive integer. Prove that two numbers na +3n+6 and n2 + 2n +7 cannot be prime at the same time. (b) Find 15261527863698656776712345678%5 without using a calculator. (c) Let a be an integer number. Suppose a%2 = 1. Find all possible values of (4a +1)%6. 2. (Integers: %, =) (a) Suppose a, b, n are integer numbers and n > 0. Prove that (a+b)%n = (a%n +B%n)%n. (b) Let a,...
1. DOES A DIGIT APPEAR IN AN INTEGER, Write a recursive function appears(n,i) that returns true if the digit i appears in the positive (base 10) integer n, false if it does not in c++. ex. Enter a positive integer: 54321 Enter a digit: 7 The digit 7 does NOT appear in the integer 54321. 2. IS AN INPUT STRING A PALINDROME? A palindrome is a string in which the letters are the same in both directions, disregarding capitalization and...
DEFINITION: For a positive integer n, τ(n) is the number of positive divisors of n and σ(n) is the sum of those divisors. 4. The goal of this problem is to prove the inequality in part (b), that o(1)+(2)+...+on) < nº for each positive integer n. The first part is a stepping-stone for that. (a) (10 points.) Fix positive integers n and k with 1 <ksn. (i) For which integers i with 1 <i<n is k a term in the...
Problem (5), 10 points Let a0:a1, a2, be a sequence of positive integers for which ao-1, and a2n2an an+ for n 2 0. Prove that an and an+l are relatively prime for every non-negative integer n. 2n+an for n >0 Problem (5), 10 points Let a0:a1, a2, be a sequence of positive integers for which ao-1, and a2n2an an+ for n 2 0. Prove that an and an+l are relatively prime for every non-negative integer n. 2n+an for n >0
Ok = (6) Let n be a positive integer. For every integer k, define the 2 x 2 matrix cos(27k/n) - sin(2nk/n) sin(2tk/n) cos(27 k/n) (a) Prove that go = I, that ok + oe for 0 < k < l< n - 1, and that Ok = Okun for all integers k. (b) Let o = 01. Prove that ok ok for all integers k. (c) Prove that {1,0,0%,...,ON-1} is a finite abelian group of order n.
16.5.3 If f(2r)2+xforall x >0, then what is 2f(x)? (Source: AHSME 16.5.4 Let () and S(n) denote the product and the sum, respectively, of the digits of the integer n. For example, P(23-6 and S(23)-5. Suppose N is a two-digit number such that N-P(N) + S(N). What is the units digit of N? (Source: AMC 12) Hints: 155 function f(x, y) of two variables has the property that 16.5.5 A fx, y)+f(x-1,x -y) 16.5.3 If f(2r)2+xforall x >0, then what...
1. Let n be a positive integer with n > 1000. Prove that n is divisible by 8 if and only if the integer formed by the last three digits of n is divisible by 8.