{-74 + 1, ..., 74 – 1}. (b) Hence, find a solution to the congruence x' = 99 mod 74.

Q: 32-2n Σ=3 Ln=3_25-n

A: We have to evaluate ∑n=3∞ 32-2n25-n.

Q: solve 5.3 (using k(pi-k)/2) and setting k =pi/2 or by your own methods if you have easier ones

A: It is known to us that

Q: Let a and b be positive integers and let b= qa +r, where 0 ≤r <a. Prove that (a) 2b - 1 = 2" - 1…

A: Let  and  be two positive integers.Let  where (a) We  have to prove that      Now, observe that…

Q: Show that if |z1| ≤ 2 and |z2| ≤ 3, then one has |(3+4i) 2₁ (1 + i)21 22 ≤ 10+6√2 21 -

A:

Q: Multiply. V3 cis (똥). 유cis () 4 o B cis (3) 3/2 8 2/3 8. cis () o 좋 cis () 3/2 8. 31π 18 2/3 cis (중)…

A:

Q: For which value(s) of k does X₁+kx₂=1 kx₁ + x₂ = 1 have infinitely many solutions?

A: Here for two given linear equation we have to find value of k such that it has infinitely many…

Q: 1. You want to prove the following formula by induction. n Σ(2i + 2) = n(n+3) i=1 (a) Complete the…

A: Given  ∑i=1n2i+2=nn+3 Basis step: Prove the result for n=1. Inductive Step: Assume the result is…

Q: (2) (a) Find all of the solutions modulo 15 to 6x = 15 mod 18. Express your answer in terms of the…

A: Given, 6x≡15 mod 18. As it is known that, an≡bn mod cn ⇔ a≡b mod c. Therefore, we have 6x≡15 mod 18…

Q: (a) Find an r E Z that makes the following congruence true: 5x = 1 (mod 24). (b) Prove that there is…

A: Solution a: Given  5x≡1 mod 24 For x=1 5≡5 (mod 24) for x=2 10≡10 (mod 24) for x=3 15≡15 (mod 24)…

Q: Prove E-1(4i – 3) = n(2n – 1) for natural numbers n 2 1. i=1

A:

Q: Prove that Σ(8i-5)=4n²-n for every positive integer n. i=1

A:

Q: Let p be prime. (a) If a 1 (mod p), show that 1+ a + ...+ aP-1 = 1 (mod p). (b) If a -1 (mod p),…

A:

Q: (a) Explicitly check that [7] + [21] = [98] + [–5] in Z13- (b) Find another value of n such that [7]…

A:

Q: 3. (i). Suppose f (x) is a polynomial with integer coefficients such that f(2) = 0(mod 5) f(4) =…

A: Given that  (a) f(2)≡0(mod 5)f(4)≡0(mod 9) We have to find a polynomial such that f(x)≡0(mod 45)…

Q: n Prove that (3i – 2) = ; (3n – 1) for all integers n > 1. - i=1

A:

Q: Use Bernoulli's inequality to prove that n + √)" = 1+ > √n for all n EN.

A: From Bernoulli inequality we know that  if x&gt;-1 then : 1+xn≥1+nx for n≤0 or n≥1. Therefore for…

Q: 8.5. Find all incongruent solutions to each of the following linear congruences. (a) 8x = 6 (mod 14)

A: