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Unformatted text preview: f. Least Common Multiple (LCM) g. Connection between LCM and GCD h. Proof that 2 is irrational. Sample Questions from Spring 2001 Final Exam 1) Use induction to prove that 64 | (3 2n 8n 1) for all integers n 0. 2) (10 pts) Let c be an integer such that 3 | c. Prove that (c+1) 3 1 (mod 9). 3) Prove the following inequality for all positive integers n: 1 2 ) 1 ( log 2 1 2 +- = n i n i n (Hint : Remember that log 2 (2 x y) x when x is a positive integer and y is a non-negative integer such that y < 2 x .) 4) (10 pts) In a gumball machine there are 32 red gumballs, 14 green gumballs, 30 white gumballs, and 5 purple gumballs. A devoted customer purchases 10 gumballs. How many combinations of gumballs can the customer receive? (Remember the order in which you receive the gumballs does not matter. Only the total set of 10 gumballs matters.) Note: Other questions will be added tomorrow....
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- Spring '09