Factor x6 +1 into irreducible polynomials mod 2.

Q: The sum of any two integers of the from 4k + 1is always: Od Zero Even prime

A: Sum of two integers of the form 4k+1

Q: 1. Prove that 7(x – 1) 2.

A: Let, f(x)= ex-7x+7 ,for x≥2. Differentiating both sides with respect to x, f'(x)= ex-7≥0 . This…

Q: Prove that 2n32n - 1 is always divisible by 17.

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Q: Use the Chinese Remainder Theorem to find a solution to the congruences ´x = 2 mod 10 x = 3 mod 1001…

A: The given system is, x≡2mod10x≡3mod1001x≡3mod9 Chinese reminder theorem: If m1,m2,…,mn are positive…

Q: Find the solution to 34x = 77(mod 89) that is positive and less than 89.

A: given that                                          34x≡77 (mod 89) we have to find positive…

Q: Show that 3 · 4n + 51 is divisible by 3 and 9 for all positive integers n.

A:

Q: Give a combinatorial interpretation of the coefficientof x6 in the expansion (1 + x + x2 + x3 +⋯)3.…

A: According to the given information, it is required to give a combinatorial interpretation of the…

Q: Prove by induction. 3F, = Fn+2+ Fn-2 for n > 2

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Q: LI) Prove that (3

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Q: 8"-3 is a multiple of 5. In other words, prove that 8"-3"=5 for some i integer j.

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Q: 2X=3 mod 5 Show your work

A: the given congruence equation is 2x≡3mod 5 the general congruence equation is ax≡bmod c…

Q: Find all integer solutions to the equation 23. + 6x2 + x + 6 =1 mod 74. Explain your method and cite…

A: en x3+6x2+x+6 = 1 mod 71

Q: Find the four solutions of x* = 1 (mod 13).

A: Since you have asked multiple questions we will solve the first question for you .To get any…

Q: Find all integers x + 3 such that x – 3 is divisible by x – 3. |

A: To find all integers such that x^3-3 is divisible by x-3

Q: Suppose f(x) is a degree-2 polynomial. How many solutions x are there to f(x) = 0 (mod 37 · 59) such…

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Q: Find x such that 1577 =x (mod11), 0sx<11. You must show all the detailed steps

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Q: (b) Since 61 is a prime with 61 = 1 (mod 12) the polynomial X? + 3 has roots in Z/61Z by (a): they…

A: Here we need to solving ....(i) knowing that .....(ii)

Q: 5. Use the method of successive squaring to compute 3875 (mod 103) 1

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Q: b. Find the remainder using Fermat's Little theorem 452 mod 11.

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Q: (a) Solve for x if 25x = 15(mod 29).

A: Linear congruence

Q: 7. Use the Chinese remainder theorem to solve the following: 3x 2 (mod4) 5x =1 (mod 7) 8x 5 (mod9) I…

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Q: Solve for x : 5x ≡ 7 (mod 6).

A: Here remainder is 7 and modulus is 6 we can write it as  5x-7 =6n          Where  is an positive…

Q: Fermat's theorem implies that for a prime p. 2P = 2( mod p). Will the converse be true that if 2P =…

A: Note: As per our company guidelines we are supposed to answer the first three sub-parts of the…

Q: find the integer x such that 0 < x < 232 and 144x = 4(mod 233). Show your work.

A:

Q: Show that 3·4n+ 51 is divisible by 3 and 9 for all positive integersn

A: Consider the provided expression,

Q: Number theory

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Q: using the Chinese Remainder Theorem how do I solve this systm of congruences?? 2x ≡ 1 (mod 5) 5x ≡ 4…

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Q: 2X+3=1 mod 5 Show your work

A: We have solve given problem:

Q: Find all integer values of x for which 3x = 4 (mod 8).

A: given congruence equation 3x≡4(mod 8)

Q: Solve x2+3x+2 is congruent to 0 (mod 55)

A: 55=5×11 and gcd(5,11)=1, hence they are co-prime.The given system of equation x2+3x+2≡0mod 55 is…

Q: Use Hensel Lifting (twice) to find a solution to x² = 7 (mod 374).

A: According to the given information, it is required to use Hensel lifting method to find the…

Q: Obtain the expansion for (3 + 2x)* in ascending powers of x

A: According to binomial expansion,  a+bxn=an+n1an-1bx+n2an-2b2x2+n3an-3b3x3+....+nn-1abn-1xn-1+bnxn…

Q: Show that if x is an integer, then gcd (x+1, x^2 -x +1) divides 3. Use this result to find knowing…

A: Solving

Q: If 2n + 1 is prime for some n ≥ 1, prove that n is a power of 2.

A: We have given 2n +1 is prime for some n ≥ 1 We have to prove that n is a power of 2.

Q: Solve by using the Chinese remainder theorem for the congruence X 2 (mod 7) X 4 (mod 5)

A: Given that This can be written as,

Q: Find [7]^-1 in Z55 under multiplication.

A:

Q: Show that the product of two numbers of the form 4n + 1 is still of that form. Hence show that there…

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Q: 2. Find an inverse of 2 modulo 17 that is positive and less than 17.

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Q: 2.x = 3 (mod 11) тod 10x = 7 (mod 13) тod

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

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Q: 8. Show that 18!= -1 (mod 437). 9. Using Euler's Theorem, find the last two digits of 44443333 10.…

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Q: Find the mod of (2-4i)^1/2

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Q: 6. Let f(x) be a polynomial with integral coefficients (like f(x) = 3x^^5-x^2+7) and suppose there…

A: Given:- fx=3x5-x2+7 There are 2 solutions  x mod 5 such that fx is congruent to 3 mod 5 and there…

Q: Let x, y ∈Z. Prove that if x≡1(mod 5) and y ≡2(mod5 ), then x^2-y^2≡0(mod 5).  Give 2 examples to…

A: Given, x,y∈ℤ and x≡1mod 5, y≡2mod 5

Q: Obtain the expansion of ; up to the term in x* 1-3x2

A: We know 11-x = 1+x+x2+x3+ ⋯

Q: Prove by induction: 1- 3h 27 3* 2

A:

Q: (b) Since 61 is a prime with 61 = 1 (mod 12) the polynomial X2 +3 has roots in Z/61Z by (a): they…

A: Given that, 61 ≡ 1 mod (12) The polynomial x2 + 3 has roots in Z/61Z. The roots are 27 + 61Z and -27…

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: Number theory

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Unitary Method

The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products. 

Speed, Time, and Distance

Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.

Profit and Loss

The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.

Units and Measurements

Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.