excercise 4.1 #24 16. Find a to the that if m is an m 2, then (a + b) mod17. Find a to the that if m is an with m 2, modf the fa154Congruencesb; (mod m).*(u pou) 'q]= 'v]] (qa) a; = b; (mod m).17. Find a counterexample to the statement that if m is an integer with m 2, then am = (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 mm) mod m for all integers a and b.%3Dpour19. Show that if m is a positive integer with m 2, then (ab) mod m = ((a mod m)(b modamod m for all integers a and b.%3DIn Exercises 20–22, construct tables for arithmetic modulo 6 using the least nonnegative residuesmodulo 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 reada) 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 andis prime?26. Show that if ak = bk (mod m) and ak+l = bk+1 (mod m), where a, b, k, and m are integerswith 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?%3D27 Show that if n is an odd positive integer, then1+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, ue13 +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 that1²+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

Question

excercise 4.1 #24 16. Find a to the that if m is an m > 2, then (a + b) mod17. Find a to the that if m is an with m > 2, modf the fa154Congruencesb; (mod m).*(u pou) 'q]= 'v]] (qa) a; = b; (mod m).17. Find a counterexample to the statement that if m is an integer with m > 2, then am = (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 mm) mod m for all integers a and b.%3Dpour19. Show that if m is a positive integer with m > 2, then (ab) mod m = ((a mod m)(b modamod m for all integers a and b.%3DIn Exercises 20–22, construct tables for arithmetic modulo 6 using the least nonnegative residuesmodulo 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 reada) 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 andis prime?26. Show that if ak = bk (mod m) and ak+l = bk+1 (mod m), where a, b, k, and m are integerswith 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?%3D27 Show that if n is an odd positive integer, then1+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, ue13 +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 that1²+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

Accepted Answer

We have to find decimal digits occur as the final digit of a fourth power of an integer.

b) 100 hours aft 24. Which decimal digits occur as the final digit of a fourth power of an integer? 25 What can you conclude if a² = h² (mod n) where a and h are integers and ni-