for all integers a and b, a² + b² = 0, 1, 2, 4 or 5 modulo 8. Deduce that there do not a and b such that a² + b² = 12345790.

Q: 2. Euclidean Algorithm Use the Euclidean algorithm to find gcd(12345, 678).

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Q: .3x = 4(mod5).

A: From the given expression of the modulus, we will have to find out the value of X. This is…

Q: Find a positive integer less than 8 for x that satisfies x = 720 + 3 (mod 7).

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Q: if a equiv b (mod n) , then a⁴ equiv b⁴ (mod n) for any positive integer 4.

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Q: The notation x = y mod n is used to indicate the unique integer x in the range 0 :$ x < n such…

A: Given statement is that the notation x = y mod n is used to indicate the unique integer x in the…

Q: b=3(mod19)

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Q: If x^677 ≡ 3 mod 2020, find x.

A: We have   x677≡3mod2020⇒x677-3=2020m   where m is integer different values of m we get different…

Q: Give a definition of an inverse of a number a modulo m.

A: if the two integer a and m modulo then  let x is an integer value so the x is inverse of an only if…

Q: Find a positive integer less than 6 for x that sat 2x + 340 = 1703 (mod 5).

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Q: For integers a and b, prove that (a + b) 3 a3 + b3 (mod 3).

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Q: Suppose that a is an odd integer and (a, 91) = 1. Prove that a12 ≡ 1 (mod 1456).

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Q: Find the least residue of  -5 -46 modulo 14.

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Q: (b) Find 95! (mod 96).

A: We have to find the value of 95!(mod 96). Since 95!=95*94*93*…*8*7*6*5*4*3*2*1.

Q: Find 1112 mod 13.

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Q: Compute (3714 - 39) mod 9 without a calculator.

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Q: Problem 7 Suppose b is any integer. Ifb mod 12 = 5, what is 6b mod 12? In other words, if division…

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Q: SorvE THモ CONGROEモNしE 626 X Rエ mod5

A: As 5 is an odd number, by fermat's theorem, we have             x5-1 = 1 (mod 5)           => x4…

Q: (b) Compute 9-¹ (mod 19) by listing the multiples. Show your work.

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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: x² + x = 13(mod17) X= * (mod17)

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Q: 0, the remainder is 2; and when divided by 11, the remainder is 0. remain is 5; when divided by

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Q: Let a and b be relatively prime positive integers. Show that a(b) + b(a) = 1(mod ab).

A: Solution:

Q: Suppose that a and b are integers with a = 11(mod 19) and b = 3(mod 19). Find the integer c such…

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Q: Find a number a such that 70a = 1 mod 17.

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

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Q: Prove that a1 + a2 = b₁ + b₂ mod n.

A: We have to prove that a1+a2=b1+b2 (mod n).Given:a1≡b1 (mod n)a2≡b2 (mod n)This means there exist…

Q: For any integer a, prove that a = 0 or 1 (mod 3).

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Q: Find the remainder of 5130 modulo 28

A: Chinese Remainder Theorem : Let n1​,n2​,n3​,...,nk​∈N , with GCD(ni​,nj​)=1 for each i=j. Let…

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

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Q: Please find x

A: To solve the congruence x173≡428(mod541), we can follow these steps:1. Check if 541 is prime: First,…

Q: 5x² + 3x = 3(mod17) X = (mod17)

A: Given that 5x2+3x≡3 (mod 17)x(5x+3)≡3 (mod 17)x≡3 (mod 17)  or   (5x+3)≡3 (mod 17)x≡3 (mod 17)  or…

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: If 3x = 6 mod 9, when -2 <x< 8. Then O x=(-1, 3, 5} x=(-1, 2, 5} O x=(-1, 2, 3} O x=(-1, 5, 6}

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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: b) Find an integer x such that 37x = 1 (mod 101).

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Q: Find all integers sx such that 2x = 5 (mod 7).

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Q: List all the * = 10.

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Q: Give all non-negative solutions for 10r =-17 mod 4.

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