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Consider Pythagorean triples (a, b, c) with a2 + b2 = c2, and with a, b, c ∈ Z. Suppose moreover that a and b have no common factors.
(a) Show that either a or b must be odd, and the other even.
(b) Show in this case (assuming a is odd and b even) that there are integers m, n so that a = m2 - n2, b = 2mn, and c = m2 + n2. [Hint: Note that b2 = (c - a)(c + a), and prove that (c - a)/2 and (c + a)/2 are relatively prime integers.]
(c) Conversely, show that whenever c is a sum of two-squares, then there exist integers a and b such that a2 + b2= c2.
For each object, its SA, and some dimensions are given. Calculate the dimension indicated by the variable.right square pyramid SA = 132 cm square and base is 6 cm on each side
The price of each unit of the good in Year 1 was $280 and it was $320 in Year 2. Suppose Year 1 is taken as the base year for the calculation of GDP.
Find the roots of the characteristic polynomial, r+ and r-, larger and smaller or equal, respectively. Find the real-valued fundamental solutions to the differential equation associated to the roots in part (a).
If Ax = b is consistent, then the solution set of Ax = b is obtained by translating the solution set of Ax = 0.
Relate the three measures of center, mean, median, and mode, with the levels of measurement, nominal, ordinal, interval, and ratio. Be sure to include which measures of center are appropriate for each level of measurement and why.
Find the exact length of the curve. use a graph to determine the parameter interval. R = cos2(θ/2)
what is the present value of R1 000 to be received in 10 year's time assuming a discount rate of 12% per annum?
What deposit made at the end of each quarter will accumulate to $28 000.00 in four years at 4% compounded quarterly? Please use PMT formula.
Show that the account grows at a rate proportional to A. (Continuous compounding means interest is being added continually, rather than after a specified length of time.)
Total number of ways a customer can place the order, A bagel shop offers 14 varieties of bagels, 11 flavors of cream cheese, and 15 flavors of coffee. How many different orders for a bagel
Problem 1: Find the average rate of change of the function over the given interval.
In our reading this week in Chapter 2, Murphy points to the Fugitive Slave Act as an example of a law that is so morally wrong that subjects would be justified in disobeying such law and in fact, it could be said that it would be the duty of citiz..
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