math135-a5

# math135-a5 - k integers could be 4 Either prove or give a...

This preview shows page 1. Sign up to view the full content.

Assignment 5 – Questions 1. Prove that any two consecutive integers are coprime. 2. Prove that the Diophantine equation ax 2 + by 2 = c does not have any integer solutions unless gcd( a,b ) | c . If gcd( a,b ) | c , does the equation always have an integer solution? 3. (a) Prove that if n 3, then n ! + 3 is not prime. (b) Prove that for every k P , k consecutive positive integers that are not prime can be found. (A good way to prove this is to explicitly show what these
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: k integers could be.) 4. Either prove or give a counterexample to, a 3 | b 3 if and only if a | b . 5. Find the prime factorization of gcd( a,b ) and of lcm( a,b ), where a = 25! and b = (5500) 3 (1001) 2 . 6. Find the number of positive integers whose prime factors are 2, 3 and 5 and which have exactly 20 positive divisors. 1...
View Full Document

## This note was uploaded on 07/24/2011 for the course MATH 135 taught by Professor Andrewchilds during the Winter '08 term at Waterloo.

Ask a homework question - tutors are online