Inverses 3. Let g: N\ { 1 } → N be defined by g(n)-the product of the distinct prime factors of n. a. Find g(2,4, 6, 8,...
Find the smallest positive integer that has precisely n distinct prime divisors. 'Distinct prime divisor'Example: the prime factorization of 8 is 2 * 2 * 2, so it has one distinct prime divisor. Another: the prime factorization of 12 is 2 * 2 * 3, so it has two distinct prime divisors. A third: 30 = 2 * 3 * 5, which gives it three distinct prime divisors. (n = 24 ⇒ 23768741896345550770650537601358310. From this you conclude that you cannot...
Question 9 (6 points) (2,4, 3) and g (1, 5, 2) are permutations defined on S -(1, 2, 3, 4, 5), What is a) (fog)(4) [What is the result of applying fo g to 4] (gof)(5) [What is the result of applying g e f to 5] b)
1.(16) Let P be an inner product space with an inner product defined as <.g > Ox)g(x)dx a) Let / =1+x.8=-2+x-x. Compute: <.8 >. The angle between / and g, and proj, b) Let h=1+ mx' in P Find m such that and h are orthogonal c) Let B = (1+x.I-XX+X' is a basis for P. Use the Gram-Schmidt process to covert B to an orthogonal basis for P. 2. Suppose and ware vectors in an inner product space V...
4x-3 The function g is defined by g(x) = x+1 Find g(n+3). 8 (n + 3) = 1 xo?
6-3. Draw the network defined by N = {1,2,3,4,5,6} A = {(1,2), (1,5), (2,3), (2,4), (3, 4), (3,5), (4,3), (4,6), (5,2), (5,6)}
Decide whether or not the matrices are inverses of éach other. and 0 1 -110 10 A]Yes' 」 B) No Find the inverse of the matrix, if it exists. 8) A36 A) B) C) D) T5亏 15 5 15可 15 3 Compute the determinant of the matrix. 2 5 5 9) -2 2 -3 4 2 -5 A)-162 B)-42 C) 42 D) 162 a b c 10) Let d ef g h i 8. Find the determinant below. a b...
0. For n E N, n > 1, let s, be defined by 8. *Let s1 1 +S2m 2 S2m-1 S2m+1 $2m 2 Find lim s, and lim s,.
Question 3 (a) Write down the prime factorization of 10!. (b) Find the number of positive integers n such that n|10! and gcd(n, 27.34.7) = 27.3.7. Justify your answer. Question 4 Let m, n E N. Prove that ged(m2, n2) = (gcd(m, n))2. Question 5 Let p and q be consecutive odd primes with p < q. Prove that (p + q) has at least three prime divisors (not necessarily distinct).
The one-to-one functions g and h are defined as follows. g={(-5, -4), (1, -6), (3, -8), (5, 3)) h(x)=2x+ 13 Find the following. ? X ()c) = The one-to-one functions g and h are defined as follows. g={(-5, -4), (1, -6), (3, -8), (5, 3)) h(x)=2x+ 13 Find the following. ? X ()c) =
1. Let ū= (2,4,-1), v = (3.-3,-1) (a) Compute: x ū (b) Compute: ü x 7 (c) Is the cross product commutative? If not, what is it instead? 2. Let A = (7, -11,3), B = (1,9, -3), C = (-6,3, -2), D= (0,-8, 12), E = (1, -13,2) (a) Give the vector equation of a line passing through the points A, B. (b) Find the equation of the plane containing the points C,D,E. (c) Find the point of intersection...