Hello Thorgerdur, You can begin both problems by replacing 314 with its modular 7 representation: 314^163 = 6^163 mod 7 (a) Note how powers of 6 repeat cyclically in the modular 7 number system: 6^1 = 6 mod 7 6^2 = 1 mod 7 6^3 = 6 mod...
Hello Thorgerdur, You can begin both problems by replacing 314 with its modular 7 representation: 314^163 = 6^163 mod 7 (a) Note how powers of 6 repeat cyclically in the modular 7 number system: 6^1 = 6 mod 7 6^2 = 1 mod 7 6^3 = 6 mod...
Hi Lily, The augmented matrix is 1 2 0 2 -1 (k-3) 3 -2 2 (k+3) (k+3) 5 Its row echelon form is 1 2 0 2 0 (k-1) 3 0 0 0 k 1 When does this system have a unique solution? Try applying the fact that Ax=b has a...
Hi Courtney, You can think of the points X, Y, Z as vectors originating from the origin. Each of these "vectors" would have a magnitude associated with it, found by using the Pythagorean Theorem. For example, the magnitude of X would be sqrt(12^2 + 9^2) =...
Hi Ana, To visualize how the graph moves, rewrite y = (x - 3)^2 + 4 so that it is easier to compare with y = x^2. old: y = x^2 new: (y' - 4) = (x' - 3)^2 Now you can see that the transformation changed y to (y' - 4) and x to (x' - 3). y = y' - 4 ---->...
Hi Brittany, The only numbers that square to one are 1 and -1. 1 = 1 x 1 1 = (-1) x (-1) So since (tan x) squares to one, (tan x) must be 1 or -1. There are two values of x in the interval [0, pi] that make it happen. tan (pi/4) = 1 tan (3*pi/4) = -1
Here is a graphical way to think of it. Suppose you wanted to divide the number 1 by 0. In other words, you want to find y = 1/x when x = 0. Well, try graphing the function y = 1/x. You will notice that to the left of x = 0, the value of y approaches negative infinity...
Hi Maria, One way to solve this problem is to use the Law of Sines: sin(A)/a = sin(B)/b = sin(C)/c. Here a, b, c are the lengths of the sides, and A, B, C are the angles across from (not touching) the sides a, b, c respectively. Since you are given b, c, and B, you could...
Hi Barb, The typical way to evaluate 67 x 436 is by multiplying and adding: (60 + 7) x 436. Usually this is written as: 436 x 67 --------- 3052 = 436 x 7 + 26160 = 436 x 60 ---------- ...
Here is a way to remember that the angles of a triangle sum to 180 degrees. Imagine a squat triangle, with two very acute angles and one very obtuse angle. If this triangle is very stout, then the acute angles are ~0 and the obtuse angle is ~180 degrees.
Let's suppose 9 is a factor of the number N. That means N is 9 times some integer M. N = 9*M Since 9 = 3*3, we can also write N as N = 3*3*M That means N is 3 times some integer (3*M). So 3 is also a factor of N.