In Exercises 20-22, construct tables for arithmetic modulo modulo 6 to represent the congruence classes. 20. Construct a table for addition modulo 6. 21. Construct a table for subtraction modulo 6. 22. Construct a table for multiplication modulo 6. 23. What time does a 12-hour clock read a) 29 hours after it reads 11 o'clock? c) 50 ho b) 100 hours after it reads 2 o'clock? 24. Which decimal digits occur as the final digit of a four
In Exercises 20-22, construct tables for arithmetic modulo modulo 6 to represent the congruence classes. 20. Construct a table for addition modulo 6. 21. Construct a table for subtraction modulo 6. 22. Construct a table for multiplication modulo 6. 23. What time does a 12-hour clock read a) 29 hours after it reads 11 o'clock? c) 50 ho b) 100 hours after it reads 2 o'clock? 24. Which decimal digits occur as the final digit of a four
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
excercise 4.1 #22
![16. Find a to the that if m is an m > 2, then (a + b) mod
17. Find a to the that if m is an with m > 2, mod
f the fa
154
Congruences
b; (mod m).
*(u pou) 'q]= 'v]] (q
a) a; = b; (mod m).
17. Find a counterexample to the statement that if m is an integer with m > 2, then a
m = (a mod m)(b mod m) for all integers a and b.
m = a mod m +b mod m for all integers a and b.
18. Show that if m is a positive integer with rn > 2, then (a+b) mod m = (a mod m
m) mod m for all integers a and b.
%3D
pour
19. Show that if m is a positive integer with m > 2, then (ab) mod m = ((a mod m)(b moda
mod m for all integers a and b.
%3D
In Exercises 20–22, construct tables for arithmetic modulo 6 using the least nonnegative residues
modulo 6 to represent the congruence classes.
20. Construct a table for addition modulo 6.
21. Construct a table for subtraction modulo 6.
22. Construct a table for multiplication modulo 6.
23. What time does a 12-hour clock read
a) 29 hours after it reads 11 o’clock?
c) 50 hours before it reads 6 o'clock?
b) 100 hours after it reads 2 o'clock?
24. Which decimal digits occur as the final digit of a fourth power of an integer?
25. What can you conclude if a² = b² (mod p), where a and b are integers and
is prime?
26. Show that if ak = bk (mod m) and ak+l = bk+1 (mod m), where a, b, k, and m are integers
with k > 0 and m > 0 such that (a, m) = 1, then a = b (mod m). If the condition (a, m) =!
is dropped, is the conclusion that a = b (mod m) still valid?
%3D
27 Show that if n is an odd positive integer, then
1+2+3+ . . + (n – 1) = 0 (mod n).
Is this statement true if n is even?
28. Show that ifn is an odd positive integer or if n is a positive integer divisible by 4, ue
13 +23 + 33 + . . .+ (n – 1) =0 (mod n).
Is this statement true if n is even but not divisible by 4?
29. For which positive integers n is it true that
1²+2²+32+... + (n – 1)? =0 (mod n)?
33. Show that if n = 3 (mod 4), then n cannot be the sum of the squares of two](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcfba8c1b-379a-495a-9284-26414a9f3892%2Fe30af42d-1bd6-42ee-b094-87cf3c2522b9%2Fen2cslp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:16. Find a to the that if m is an m > 2, then (a + b) mod
17. Find a to the that if m is an with m > 2, mod
f the fa
154
Congruences
b; (mod m).
*(u pou) 'q]= 'v]] (q
a) a; = b; (mod m).
17. Find a counterexample to the statement that if m is an integer with m > 2, then a
m = (a mod m)(b mod m) for all integers a and b.
m = a mod m +b mod m for all integers a and b.
18. Show that if m is a positive integer with rn > 2, then (a+b) mod m = (a mod m
m) mod m for all integers a and b.
%3D
pour
19. Show that if m is a positive integer with m > 2, then (ab) mod m = ((a mod m)(b moda
mod m for all integers a and b.
%3D
In Exercises 20–22, construct tables for arithmetic modulo 6 using the least nonnegative residues
modulo 6 to represent the congruence classes.
20. Construct a table for addition modulo 6.
21. Construct a table for subtraction modulo 6.
22. Construct a table for multiplication modulo 6.
23. What time does a 12-hour clock read
a) 29 hours after it reads 11 o’clock?
c) 50 hours before it reads 6 o'clock?
b) 100 hours after it reads 2 o'clock?
24. Which decimal digits occur as the final digit of a fourth power of an integer?
25. What can you conclude if a² = b² (mod p), where a and b are integers and
is prime?
26. Show that if ak = bk (mod m) and ak+l = bk+1 (mod m), where a, b, k, and m are integers
with k > 0 and m > 0 such that (a, m) = 1, then a = b (mod m). If the condition (a, m) =!
is dropped, is the conclusion that a = b (mod m) still valid?
%3D
27 Show that if n is an odd positive integer, then
1+2+3+ . . + (n – 1) = 0 (mod n).
Is this statement true if n is even?
28. Show that ifn is an odd positive integer or if n is a positive integer divisible by 4, ue
13 +23 + 33 + . . .+ (n – 1) =0 (mod n).
Is this statement true if n is even but not divisible by 4?
29. For which positive integers n is it true that
1²+2²+32+... + (n – 1)? =0 (mod n)?
33. Show that if n = 3 (mod 4), then n cannot be the sum of the squares of two
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