17. For all integers a, if a mod 6 = 3, then a mod 3 + 2. %3D

Q: 9. Consider modulo 11. a. Find 35 mod(11). b. Find a negative integer x such that x mod(11) is…

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Q: If m is an integer, and if (m mod 11) = 6, then what is (5m mod 11)? 2 4 11

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Q: If a = b (mod m) and a = b (mod n) then a = b (mod mn)

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Q: If a= b (mod n) and gcd (a, n) = d, then gcd (b, n) = d :Select one True False

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Q: (a) Show that if n 7 mod 8, then n + 6n 3 mod 8.

A: If, n = 7 (mod 8) So, n² = 7² (mod 8) So, n² = 49 (mod 8) So, n² = 1 (mod 8)

Q: Suppose a and b are integers, a = 15 (mod 23), andb=7 (mod 23). Then integer c with Osc< 22 such…

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Q: 12. Prove that if gcd(a, n) { b, then ax = b (mod n) has no solutions.

A: To prove the given statement, it is equivalent to prove its contrapositive statement which is "if…

Q: 2. If a = 1 (mod p), then a=1 (mod p). Select one: True False

A: If , then . The truthfulness of the given statement can be proved by taking an arbitrary value for…

Q: Suppose b is any integer., If b mod 12 = 7, what is 4b mod 12? In other words, if division of b by…

A: Given : b mod 12 = 7

Q: 5- If a = b mod(n), then ac = bc mod(n). 6- If a b mod (n), then ak = b*mod(n) for every positive…

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Q: 5- If a = b mod (n), then ac = bc mod (n). 6- If a = b mod(n), then ak = bkmod(n) for every positive…

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Q: For which primes p > 5 is -5 a square mod p? Your answer should be of the formp = {a1, a2, ..., ak}…

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Q: 15 Composite Then and O modn Show that if n there are with X#0 od n and y7 but where yy= Omod n.

A: Given n is a composite number

Q: Suppose b is any integer. If  b mod 12 = 5,  what is 6b mod 12? In other words, if division of b by…

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Q: Show that if a ≡ b (mod m) and c ≡ d (mod m), then a + c ≡ b + d (mod m).

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Q: 9 The residue of 6 (mod 5) is (Round to the nearest whole number.)

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Q: 22. Suppose c is an integer. If c mod 15 = 3, what is 10c mod 15? In other words, if division of c…

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Q: #5. Suppose that a = 2 (mod 113), and b = 5 (mod 113) What is (a b + 56b + 155) mod 113?

A: given a≡2(mod113),b=5(mod113)a3≡23(mod113)a3b≡235(mod113)56b≡280(mod113)56b≡54(mod113)155≡15mod113)

Q: Suppose c is an integer. If c mod 15 = 3, what is 10c mod 15? In other words, if the division of c…

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Q: The last two digits of 52044 is 44. If ab = 0 (mod 3), then a = 0 (mod 3) or b = 0 (mod 3). F 10. F…

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Q: Find the smallest positive integer for which x mod 3=2 and x mod 4=3   What is the next smallest…

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Q: If a = 1 (mod n) and b = 1 (mod n) then a - b = 0 (mod n) اختر واحدا: 2 3

A: True.

Q: Which of the following are true modulo 12? (1) Does 5z = 6 mod 12 have a solution? How many? (2)…

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Q: (18) If a, be Z, then (a + b)3 = a3 + b3 (mod 3).

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Q: Number theory

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Q: If n is an integer, then (2n² + 3n)# 3(mod5).

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Q: (c) For all integers a and b, if a · b = 1 (mod 6), then a =1 (mod 6) or b = 1 (mod 6). (d) For all…

A: (c) The statement is false. For example take a=5, b=5 ab=2525≡1 mod(6) Hence ab≡1 mod(6) But a≡5…

Q: Ha = b (mod n) and gcd (a,n)= d, then gcd (b,n) %3D

A: Given a≡b(mod n). This can be written as a-b≡0(mod n). Implies a-b=kn for some natural number k.…

Q: Show that y = x (mod p) leads to -y = -x (mod p).

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Q: Suppose that a and b are integers, a = 4(mod 13), and b = 9(mod 13). Find the integer c with 0<c< 12…

A: We will use the basic properties of congruence modulo n relation to answer this question.

Q: Suppose that each of a, b, and c is an integer and m is a positive integer. Show that if a = b(mod…

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Q: Suppose that a and b are integers, c b = 3 (mod 19). Find the integer c that a) с%3D 13а (mod 19).…

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Q: (4) Let p be a prime number. Then r = 1 mod p has exactly two solutions modulo p.

A: 4) To prove, x2≡1mod p has exactly 2 roots such that p is a prime number.

Q: State whether True or False: If A and B are integers and A ≡ 3 (mod 3) and B ≡ 2 (mod 6) then A+B ≡…

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Q: 23. Let a,be Z and n eN. If a =b (mod n), then a = ab (mod n). (24, If a b (mod n) and c=d (mod n),…

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Q: 4.14. Let a, b, n E Z, where n > 2. Prove that if a = b (mod n), then a2 = b (mod n). 4.15. Let a,…

A: The detailed solution is as follows below:

Q: Suppose that a and b are integers, a = 4 (mod 13), and b = 9 (mod 13). Find the integer c with 0 < c…

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Q: 29. How many different values are there for 4n(mod 7), as n ranges over all positive integers? 30.…

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Q: Suppose b is any integer. If b mod 12 = 7, what is 6b mod 12? In other words, if division of b by 12…

A: To determine the value of 6b mod(12) if b mod(12) = 7.

Q: 12. Prove that if gcd(a, n) b, then ax = b (mod n) has no solutions.

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Q: 3. If x mod 4 = 1 and y mod 4 = 3, then xy could be O 901 O 299 O 508 O 113

A: We shall use the division algorithm

Q: If ab = 0 (mod m), then either a = 0 (mod m) or D=0 (mod m). Select one: O True False

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Q: If a # 0 (mod m) and b # 0 (mod m), then ab# 0 (mod m). Select one: O True False

A: justification is given in step 2)

Q: 12.ifa b(mod n) and b a c(mod n) then ab m be(mod n)

A: We can proove in the following way:-

Q: 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).

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

Q: 20. If a e Z and a = 1 (mod 5), then a² = 1 (mod 5). If a - h (mod n) then a3 = h3 (mod n)

A: Notation: a|b  means a divides b or                   a|b means b is divisible by a.

Q: For which primes p > 7 is 7 a square mod p? Your answer should be of the form p = {a1, a2,... , a}…

A: Given: p&gt;7 ,If 7 is a square mod p.

Q: (2) If a = b modn and a = b mod m and ged(m, n): 1. then a =b mod mn.

A: The greatest common divisor of two or more integers, which are not all zero, is the largest positive…

Q: For any integers m and n, if m mod 11 is 6 and n mod 11 is 7, what is m+n mod 11? 13 9 42

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