Take non-zero positive integers m and n. Show that Z/mZx Z/nZZ/mnZ if and only if m and n are relatively prime, that is, gcd(m, n) = 1.

Q: Find all values of z for which z* + 2 – i2v3 = 0 and represent them geometrically.

A:

Q: Show that Sn is solvable when n ≤ 4.

A: Given: Sn. We have to show that Sn is solvable when n≤4.

Q: Suppose X and Y have joint characteristic function (s, t). Show that, subject to the appropriate…

A:

Q: Suppose that x and y are real numbers such that y is 6 greater than x. What is the smallest possible…

A:

Q: If n^2+6n+11 = x^2+y^2 for positive integer n, x and y prove [xy/2] is composite. Note: [t]…

A: Given :              n2+6n+11 = x2+y2              t ≤ t

Q: Show that if x, y,z are positive integers then: z-gcd (x,y) = gcd(zx ,zy)

A: The greatest common divisor (gcd) is the largest positive integer that divides each of the integers…

Q: Determine the first 3 terms in the Maclaurin expansion of y = esin², X + X + x².

A:

Q: Determine the first 3 terms in the Maclaurin expansion of y = esin² T + X + x²

A:

Q: Prove that πd/n n = nτ(n)/2.

A: your question has a silly mistake, it should be πd|n d = nτ(n)/2. supposeτ(n)=r , where r be the…

Q: Let n be a positive integer. The coefficient of x°y° in the expansion of (1+ xy+ y²)' equals

A:

Q: Is Q closed over multiplication? (Y/N)

A:

Q: 4. If x = cose + isin0 & 1 + V1– a² = na then prove that 1+ acose =(1+ nx)(1+) %3D 2n

A: Given, x= cos θ+i sinθ and na=1+1-a2      To prove:  1+a cosθ=a2n(1+nx)(1+nx)       Let  x=…

Q: 7. Find a pair of integers (x,y) such that 4021x + 2014y= 1.

A:

Q: Find the Re and Im parts of exp(z?)

A:

Q: Write a in the form a=a+T+aNN at the given value of t without finding T and N. 4 r(t) = (41²) ₁ +…

A:

Q: xʻy Let f (x, y) = if (x, y) # (0,0) and f (0,0) = 0. (x4+6y8) r4.

A:

Q: Find all gcd’s of the two Gaussian integers 5 + 3i and −5 + i.

A: By repeated use of Euclidean division algorithm  we get the value of g.c.d.

Q: Find all monic irreducible polynomials of degree 2 over Z

A: A polynomial of degree 2 in Z3 [x] is irreducible if and only if it has no roots in Z3.

Q: Prove that (1,1) is an element of largest order in Zn1 + Zn2. State the general case.…

A: We know that order of any group G×H is lcm(|G|, |H|)     |G| = order of group G. Similar for H. Now,…

Q: Indicate the least integer n such that f(x) = (3æ³ + 2x²)(5æ5 + 4æ*) € O(x").

A: Finding f(x) explicitly using multiplication of polynomial. The highest power of x or degree of f…

Q: Suppose X~N(112,16), and Z~N(0,1). Then x = 112, is equivalent to z = ______.

A:

Q: Take non-zero positive integers m and n. Show that Z/mZx Z/nZ≈ Z/mnZ if and only if m and n are…

A:

Q: Determine the first 3 terms in the Maclaurin expansion of esin² a y = + X + x².

A:

Q: Q-12)-for the model below Max z=10X1+5X2 st 4X1+2X2540 3X1+3X25 30 X12 0, X2 20 MAX Z=- O 75 O 100…

A: Linear programming. Graphical method

Q: 5. Show that for Chebyshev polynomials (Tn, Tm) = 0, n #m, π, EIN n=m= 0, n = m = 0.

A: We have to solve given problems;

Q: Find a number z (except 0) such that Az³ + Bz²+ Cz = D 2 1 3 0 -2 -4 00 ^-[ ] ³-R2 :]-[+]-[1] B= 1 2…

A: Solution

Q: E Show that x'+ 3x+ 2 has 4 zerosin Ze

A: