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## Emma D.'s Resources

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.