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ac bc (mod mc). 14. Show that if a, b, and c are integers with c>0 such that a = b(mod c), then (a, c) = (b, c). II where m is a positive integer and aj, bj.

ac bc (mod mc). 14. Show that if a, b, and c are integers with c>0 such that a = b(mod c), then (a, c) = (b, c). II where m is a positive integer and aj, bj.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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excercise 4.1 #14

4.1 Introduction to Congruences 153 each requiring O((log, m)²) bit operations. Therefore, a total of O((log2 m)- log2 N) bit operations is needed. 4.1 EXERCISES 1. Show that each of the following congruences holds. a) 13 = 1 (mod 2) g) 111 =-9 (mod 40) (7 pou) = 69 (P h) 666 = 0 (mod 37) b) 22 = 7 (mod 5) e) -2 = 1 (mod 3) c) 91 = 0 (mod 13) f) -3= 30 (mod 11) 2. For each of these pairs of integers, determine whether they are congruent modulo 7. a) 1, 15 c) 2, 99 e) –9, 5 b) 0, 42 d) –1, 8 669 - 3. For which positive integers m is each of the following statements true? a) 27 = 5 (mod m) b) 1000 = 1 (mod m) c) 1331 = 0 (mod m) 4. Show that if a is an even integer, then a² = 0 (mod 4), and if a is an odd integer, then a? = 1 (mod 4). 5. Show that if a is an odd integer, then a² = 1 (mod 8). 6. Find the least nonnegative residue modulo 13 of each of the following integers. a) 22 e) – 100 8 b) 100 d) – 1 7. Find the least nonnegative residue modulo 28 of each of the following integers. c) 12,345 25 d) – 1 66 e f) –54,321 v? 22 8. Find the least positive residue of 1!+2! + 3! + · . .+ 10! modulo each of the following integers. a) 3 b) 11 c) 4 d) 23 9. Find the least positive residue of 1!+2!+ 3! + . ..+100! modulo each of the following integers. b) 7 c) 12 d) 25 a) 2 10. Show that if a, b, and m are integers with m >0 and a = b (mod m), then a mod m = b mod m. 11. Show that if a, b, and m are integers with m > 0 and a mod m = b mod m, then a = b (mod m). 12. Show that if a, b, m, andn are integers such that m > 0, n > 0, n | m, and a = b (mod m), then a = b (mod n). 13. Show that if a, b, c, and m are integers such that c > 0, m > 0, and a = b (mod m), then ac = bc (mod mc). 14. Show that if a, b, and c are integers with c>0 such that a = b(mod c), then (a, c) = (b, c). 15. Show that if a; = b; (mod m) for j= 1, 2, j = 1, 2, .. ., n, are integers, then %3D .. .,n, where m is a positive integer and a ;, b;,
Transcribed Image Text:4.1 Introduction to Congruences 153 each requiring O((log, m)²) bit operations. Therefore, a total of O((log2 m)- log2 N) bit operations is needed. 4.1 EXERCISES 1. Show that each of the following congruences holds. a) 13 = 1 (mod 2) g) 111 =-9 (mod 40) (7 pou) = 69 (P h) 666 = 0 (mod 37) b) 22 = 7 (mod 5) e) -2 = 1 (mod 3) c) 91 = 0 (mod 13) f) -3= 30 (mod 11) 2. For each of these pairs of integers, determine whether they are congruent modulo 7. a) 1, 15 c) 2, 99 e) –9, 5 b) 0, 42 d) –1, 8 669 - 3. For which positive integers m is each of the following statements true? a) 27 = 5 (mod m) b) 1000 = 1 (mod m) c) 1331 = 0 (mod m) 4. Show that if a is an even integer, then a² = 0 (mod 4), and if a is an odd integer, then a? = 1 (mod 4). 5. Show that if a is an odd integer, then a² = 1 (mod 8). 6. Find the least nonnegative residue modulo 13 of each of the following integers. a) 22 e) – 100 8 b) 100 d) – 1 7. Find the least nonnegative residue modulo 28 of each of the following integers. c) 12,345 25 d) – 1 66 e f) –54,321 v? 22 8. Find the least positive residue of 1!+2! + 3! + · . .+ 10! modulo each of the following integers. a) 3 b) 11 c) 4 d) 23 9. Find the least positive residue of 1!+2!+ 3! + . ..+100! modulo each of the following integers. b) 7 c) 12 d) 25 a) 2 10. Show that if a, b, and m are integers with m >0 and a = b (mod m), then a mod m = b mod m. 11. Show that if a, b, and m are integers with m > 0 and a mod m = b mod m, then a = b (mod m). 12. Show that if a, b, m, andn are integers such that m > 0, n > 0, n | m, and a = b (mod m), then a = b (mod n). 13. Show that if a, b, c, and m are integers such that c > 0, m > 0, and a = b (mod m), then ac = bc (mod mc). 14. Show that if a, b, and c are integers with c>0 such that a = b(mod c), then (a, c) = (b, c). 15. Show that if a; = b; (mod m) for j= 1, 2, j = 1, 2, .. ., n, are integers, then %3D .. .,n, where m is a positive integer and a ;, b;,
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