Learning Packet 4 | Number Theory PDF

Learning Packet 4| Number Theory 76 i Assess Your Understanding Concepts and Vocabulary 1. Mark TRUE or FALSE a. Every integer divisible by 5 is odd. b. Every i...

Learning Packet 4| Number Theory 76 i Assess Your Understanding Concepts and Vocabulary 1. Mark TRUE or FALSE a. Every integer divisible by 5 is odd. b. Every integer divisible by 11 is odd. c. 11 is a palindrome. d. Every palindrome is divisible by 9. e. 101000 −1 is divisible by 9. 2. Rewrite each sentence using the congruence symbol. a. n is an odd integer. b. n is an even integer. c. n is divisible by 5. d. The product of any three consecutive integers is divisible by 6. Skill Building (Problem Set) 3. Which of the following numbers are divisible by 2? By 4? By 8? a. 427,364 b. 800,358,816 4. Which of the following numbers are divisible by 3? By 9? a. 205,876 b. 5,588,610,911 5. Determine whether each number is divisible by 6. a. 87,654 b. 639,576 6. Determine whether each number is divisible by 11. a. 43,979 b. 502,458 7. Find the day of the week in each case. a. 234 days from Monday. b. 1776 days from Wednesday. 8. January 1, 2000, falls on a Saturday. What day of the week will January 1, 2020, be? (Hint: Look for leap years.) Applications and Extensions (Problem Set) 9. Using casting out nines, identify each computation as probably correct or definitely wrong. a. b. 10. Using casting out nines, find the missing nonzero digit d in each computation. a. 7961− 1976 = 59d5 b. 253 · 86 = 2d758 11. Find all four-digit integers of the form 4ab8 that are divisible by 2, 3, 4, 6, 8, and 9. 12. Find the digital root of 17761776. 13. Determine whether 61,194,858,376 can be a square. 14. Let S = {true, false}. Define a boolean function f :N→S by f (n) = true if year n is a leap year and false otherwise. Find f (n) for each year n. a. 1996 b. 2076 15. Determine the day of the week of each historical date. a. March 7, 1876 (first telephone patent issued to Alexander Graham Bell) b. December 17, 1903 (world’s first flight) c. April 12, 1961 (first human travel in space) 16. The day of January 1 of any year y can be determined using the formula JGV Pedraza Learning Packet 4| Number Theory 77 i where 0 ≤ x ≤ 6. (G. L. Ritter, 1977) Using this formula, determine the first day in each year. a. 2076 b. 3000 17. (Easter Sunday) The date for Easter Sunday in any year y can be computed as follows. Let a = y mod 19, b = y mod 4, c = y mod 7, d = (19a + 24) mod 30, e = (2b + 4c + 6d + 5) mod 7, and r = 22 + d + e. If r ≤ 31, then Easter Sunday is March r; otherwise, it is April [r (mod 31)]. Compute the date for Easter Sunday in each year. a. 1996 b. 2076 18. Solve the recurrence relation gn = gn−1 + (n − 1), where g1 = 0. 19. Develop a round-robin tournament schedule with a. Six teams b. Nine teams Explaining Concepts: Discussion and Writing (Recitation) 20. We would like to schedule a round-robin tournament with seven teams, 1 through 7. Pair team i with team j in round k, where j ≡ i+1+k (mod 7).Will this pairing provide a conflict- free and duplication-free schedule? If not, explain why. JGV Pedraza