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1. Show that there do not exist integers x and y for which 110x + 315y = 12.
2. If a and b are odd integers, prove that a2 +b2 is divisible by 2 but is NOT divisible by 4. Hint:
Any odd number can be written 2k + 1 for some integer k.
3. Prove that for any prime number p, √ p is irrational. Hint: Follow the same method that was used to prove √ 2 is irrational, by first supposing √ p can be expressed as a=b.
(a) Determine the matrix that first rotates a two-dimensional vector 180° anticlockwise, and then per- forms a horizontal compression of the resulting vector by a factor 1/2 (leavi
SOLVE THE inequation 0>-5 -X AND X Belongs TO R .Represent THE SOLUTION SET ON THE NUMBER LINE
Vector Form of the Equation of a Line We have, → r = → r 0 + t → v = (x 0 ,y 0 ,z 0 ) + t (a, b, c) This is known as the vector form of the equation of a line. The lo
how do you solve a homogeneous ode that''s not in a multiplication or division form
Bill traveled 117 miles in 2.25 hours. What was his average speed? Use the formula d = rt (distance = rate × time). Substitute 117 miles for d. Substitute 2.25 hours for t and
Give the Definition of Logarithms ? A logarithm to the base a of a number x is the power to which a is raised to get x. In equation format: If x = ay, then log a x = y.
Sales increased through only 1/2% last month. If the sales from the previous month were $152,850, what were last month's sales? Multiply through the decimal equivalent of 1/2 %
Probability Rules A probability is a number assigned to the occurrence of an event in a sample space. Probability measures must satisfy three rules. If A is an even
Rate of Change : The first interpretation of derivative is rate of change. It was not the primary problem which we looked at in the limit chapter, however it is the most signific
Consider the Solow growth model as given in the lecture notes using the Cobb-Douglas production function Y t = AK 1-α t L α t a) Set up the underlying nonlinear differen
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