9. Prove that if n = 3 (mod 4), then the equation x² + y² = n has no integer solution.

Q: How do I prove that if a, b, and c are integers with c not equal to 0 then ac = bc => a = b?

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Q: quadratic equation r2 + 4x-3 0 have a solution modulo 41"

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Q: Solve the quadratic congruence, 6x²-7x+2 ≡ 0 mod 31.

A: We have to solve the above quadratic congurence.

Q: Calculate C(12,5) by hand. As C(n,r) = n!/(n-r)!r! identify n and r, and write out every number in…

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Q: Q16

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Q: Q2: Find the solution of 143x³ + 22x¹ + 5x² + 3x + 1 = 0 (mod 11), if any.

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Q: Solve the following system 3x + 5y − 7z ≡ 8 (mod 83) 8x − 9y + 13z ≡ 13 (mod 83) 7x + 4y + 5z ≡ 15…

A: Let AX=B where, A=35-78-913745, X=xyz, B=81315 Therefore, we have: AA-1·X=A-1B           IX=A-1B…

Q: 7x = 4 (mod 12).

A:

Q: Prove that the last two digits of 2" cannot be 02 and the last three digits cannot be 108. Find all…

A: All these problems can be solved using modular arithmetic. We presented the detailed solution in the…

Q: Does x2 +3x+7=0 mod 31 have solutions? IF

A: This follows in the same way as we study quadratic equations. Here we use facts about quadratic…

Q: Find the solutions of a3 – 24.x2 + 54 = 0( mod 25).

A: want to find solution of x3-24x2+54≡0(mod 25)note that1 a≡b(mod m) f and only if m|a-b(i.e a-b=mk…

Q: Find the solutions of the congruence 15x2 + 19x ≡ 5 (mod 16).

A: Congruence modulo relation on integers:Let a,b∈Z and let m be a fixed positive integer. Then we say…

Q: (3) If today is Thursday what day of the week will it be 31 days from today? (4) If it is currently…

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Q: Let A = 2 21 -1 1 1 0 3 3 ---B 3 Calculate A¹, and use A to solve Ax

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Q: 6. Solve the equation x + 3x + 17 = 0 (mod 21)

A: The Chinese reminder theorem:                    Let p and q are coprime. Then the system equations…

Q: 7. For n ≥ 1, verify that 12+32 +52 ++ (2n- +3²+5² · + 2n-1)² = (2n+1) 3

A: Ans : page 1:Ans page 2:

Q: Find the solutions of the congruence 15x2 + 19x ≡ 5 (mod 65)

A: The given equation is: 15x2+19x≡5(mod 65)

Q: . Find all the solutions of 3x² + 2x + 1 8 (mod 63).

A: Given a quadratic congruence in the form:Ax2+Bx+C=0(mod n)multiply the above equation by…

Q: How many solutions are there to the equation X₁ + x₂ + x3 = 21, if each x; is a positive integer?…

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Q: Demonstrate that the equation x10 + y10 - z10 = 5 has no integer solution. (work with modulo 11)

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Q: Find all the solutions to the linear congruences  (b) 128x = 833 mod 1001 (c) 36x = 30 mod 42

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Q: How many positive integer solutions are there to x1 + x2 + x3 =15 ?

A: From the given information we want to find How many positive integer solutions are there to x1 + x2…

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

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Q: Prove (2b +1 , 7b + 3)=1

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Q: Show that 2x = 1 (Mod 10) has NO solutions. Show work

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Q: 7) Find HJ 2x + 25 G *+ 27 6. *+ 24

A: Given HI = 2x + 25 GJ = x +27 GI = 6 HJ = x + 24

Q: Use the baby step, giant step method to solve 2k = 7 mod 13

A: The objective is to find the value for k from the given equation 2k=7mod13 using baby step method…

Q: Find the solution to 2x = 7(mod 17) that is positive and less than 17.

A: We find solution of given congruence relation

Q: Solve 3x≡2, (mod 7) or show that no solution exists.

A: We have to solve given problem:

Q: 8. Show that the equation x³ +7y² = 3 has no integer solution. (Hint: prove by contradiction, and…

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