# hw4 - n(Hint For consecutive integers m and m 1 what does...

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CMPSC 360 Discrete Mathematics for Computer Science Fall 2009 Penn State University Assignment #4 Due: 21 September 1. If n = 3 k + 2 does 3 divide n 2 - 1? Explain. 2. Is the sum of any 5 consecutive positive integers divisible by 5? Explain. 3. True or False: A sufficient condition of a number to be divisible by 3 is that it be divisible by 9. Explain. 4. True or False: for all integers a, b, c , if a | bc then either a | b or a | c . Explain. 5. Prove that for all integers m , m 2 + 2 m - 1 is odd. 6. Prove that if n 3 is even then n is even, for all integers n . 7. Use the quotient-remainder theorem with d = 3 to prove that the product of any two consec- utive integers has the form 3 n or 3 n + 2 for some integer
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Unformatted text preview: n . (Hint: For consecutive integers m and m + 1, what does the theorem say about the possible forms of m (and hence m + 1)?) 8. Prove that if m â‰¡ n ( modd ) and n â‰¡ p ( modd ) then m â‰¡ p ( modd ). 9. Prove the following statement: for all integers n,d with d 6 = 0 d | n iï¬€ n = b n/d c Â· d 10. Prove the following statement by contradiction: There is no least rational number. 11. Prove the following statement by contraposition If the average of two numbers is greater than or equal to one of the numbers. 12. Prove that 3 âˆš 2 is irrational. (Hint: Problem 6 might be useful.)...
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