excercise 4.1 #4 4.1 Introduction to Congruences153each requiring O((log, m)²) bit operations. Therefore, a total of O((log2 m)- log2 N)bit operations is needed.4.1 EXERCISES1. Show that each of the following congruences holds.a) 13 = 1 (mod 2)g) 111 =-9 (mod 40)(7 pou) = 69 (Ph) 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, 15c) 2, 99e) –9, 5b) 0, 42d) –1, 8669 -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, thena? = 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) 22e) – 100 8b) 100d) – 17. Find the least nonnegative residue modulo 28 of each of the following integers.c) 12,345 25d) – 166 ef) –54,321v?228. Find the least positive residue of 1!+2! + 3! + · . .+ 10! modulo each of the followingintegers.a) 3b) 11c) 4d) 239. Find the least positive residue of 1!+2!+ 3! + . ..+100! modulo each of the followingintegers.b) 7c) 12d) 25a) 210. Show that if a, b, and m are integers with m 0 and a = b (mod m), then a mod m = bmod 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), thenac = bc (mod mc).14. Show that if a, b, and c are integers with c0 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;,

Question

excercise 4.1 #4 4.1 Introduction to Congruences153each requiring O((log, m)²) bit operations. Therefore, a total of O((log2 m)- log2 N)bit operations is needed.4.1 EXERCISES1. Show that each of the following congruences holds.a) 13 = 1 (mod 2)g) 111 =-9 (mod 40)(7 pou) = 69 (Ph) 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, 15c) 2, 99e) –9, 5b) 0, 42d) –1, 8669 -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, thena? = 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) 22e) – 100 8b) 100d) – 17. Find the least nonnegative residue modulo 28 of each of the following integers.c) 12,345 25d) – 166 ef) –54,321v?228. Find the least positive residue of 1!+2! + 3! + · . .+ 10! modulo each of the followingintegers.a) 3b) 11c) 4d) 239. Find the least positive residue of 1!+2!+ 3! + . ..+100! modulo each of the followingintegers.b) 7c) 12d) 25a) 210. Show that if a, b, and m are integers with m >0 and a = b (mod m), then a mod m = bmod 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), thenac = 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;,

Accepted Answer

Suppose a is an even integer
Then a=2m for some integer m
a2=(2m)2
a2=4m2Clearly 4|4so 4|4m2so…

Show that if a is an even integer, then a =0 (mod 4), and if a is an odd integer, then a =1 (mod 4).