Exercise 1. Suppose that a and b are integers, a = 3(mod11) and b = 8 (mod11) and c = a3 - b3. Then a. C=8 (mod11) b. c= 9 (mod11) c. C 10 (mod11) d. None of these

Q: - Calculate the orders of 2 (mod 67) and of 2 (mod 73).

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Q: Prove for any integers a and b, if (a3+b3) mod 3 is not 0, then (a+b) mod 3 is not 0. (Hint: recall…

A: Congruence Modulo Relation on Integers: Let a, b∈ℤ and let m be a fixed positive integer.  Then a…

Q: Express each nonnegative integer a less than 15 as a pair (a mod 3, a mod 5).

A: The values of a could be, 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14.

Q: If a = b (mod n) and a = c (mod n) then which of the following is true. (a) b = c (mod n) (b) b2 =…

A: We use the properties of congruences to answer the given question.

Q: 4. For any integer n, show that if n³ = 1 (mod 47), then n = 1 (mod 47).

A: To show that if n3≡1(mod47), then n≡1(mod47), it can use proof by contradiction.Assume that…

Q: Perform the following calculations. a. (14 + 30) mod 19 b. (14-30) mod 19 C. (3*13) mod 19

A: To perform the given calculations we shall use the following rules…

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: (c) For each integer a, if a 4 (mod 8), then a 2 (mod 8).

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Q: V. Suppose that a and b are integers, a = 7 (mod 19), and b = 5 (mod 19). Find the integer c with 0…

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Q: 98! =... .(mod101) (а) 60 (b) 50 (c) 51 (d) None of these Select one: • d x

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Q: 5. Let a, b, c, and n be integers. Prove that if a = b (mod n) and c = d (mod n), then a − c = b -d…

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Q: Which a. 24 b. 56 of the following is true? 3(mod4) 4(mod5) c. 62 = 4(mod4) d. 58 3(mod5)

A: We want to find true option.

Q: If a =b (mod m) and a = b (mod n) then a = b (mod m+n) Select one: O True O False

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Q: If a ≡ b (mod n1) and a ≡ b (mod n2), must it be true that a ≡ b (mod n1 + n2)? Either give a proof…

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Q: Show that for every integer n, either 3 | n or n² = 1 (mod 3).

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Q: Prove the following equations by induction. In each case, n is a positive integer. a. 1+4+7++ (3n-2)…

A: We are to prove   for all positive integers.

Q: Which of the following is true? a. 24 = 3 (mod4) b. 56= 4(mod5) c. 62 = 4(mod4) d. 58 = 3 (mod5) Oa…

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Q: Determine which of the following statements are true: a. 54=72(mod16)

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Q: 5. Let n be a positive integer, and assume that a and b are integers such that a = b (mod n). Prove…

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Q: If a = b (mod n) and b =c (mod n) then which of the following is false. a = c (mod n) a = c3(mod n)…

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Q: Let (a,b)=1 with a = c (mod b) and bª = c (mod a) for some integer c. Prove that ab+(b). +ba+(a) = c…

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Q: Informal Exercise 20. Observe that 4| 102. Show that to find Rem(n, 4), you just need to replace n e…

A: We need to find the remainder when n is divided by 4 .

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 that a ≡ 9(mod 13) and b ≡ 7(mod 13). Find the integer cwith0≤c≤12suchthatc≡2a2 −6b2(mod13).

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Q: 2. Prove that if a = b( mod r) and c = d( mod r) then a – c= b – d( mod r)

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Q: Recall that a = b( mod n) if a = b+ kn for some integer k. Use this to prove that if a = b ( mod n)…

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Q: V. Suppose that a and b are integers, a = 7 (mod 19), and b = 5 (mod 19). Find the integer c with 0…

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Q: If a mod b = 0, then b | a true or false

A: It's  pronounced “a is congruent to b mod n” (or “a is congruent to b modulo n”). Basically, it…

Q: For each pair of integers m and d, calculate m mod d. Show your work! a. m = 43 and d = 9 b. m 50…

A: If m and d be two integers . Then m mod d means that the remainder when m is divided by d .

Q: 999 3. Calculate (4) mod 3.

A: This simple result follows by noting that 4 is 1 mod 3. The rest of the details are given in the…

Q: Ti If a = b (mod n) and b =c (mod n) then which of the following is true. (a) a = c (mod n) (b) a2 =…

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Q: 10. Prove that a³ =a (mod 6) for every integer a.

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Q: Does 5x = 6 mod 12 have a solution? How many?

A: As per our guidelines we are supposed to answer only One question  if there are multiple questions…

Q: If a = b (mod n) and b = c (mod n) then which of the following is false. a =c (mod n) a = c (mod n)…

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Q: Suppose that a, b are integers such that 1 = -19a + 45b. Use this information to find all integers a…

A: The given equation 1=−19a+45b suggests that 1 is a linear comb.−19 and 45 with coefficients a and…

Q: If a = b (mod n) and a = c (mod n) then which of the following is true. (a) b = c (mod n) (b) b2 =…

A: We know that the property of congruence

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.