# Find the greatest common divisor (a, b) and integers m and n such that (a, b) = am + bn. a = 65, b = -91 a.) Explain in your own words what this problem is asking. b.) Explain the meaning of any notation used in the problem and in your solution. c.) Describe the mathematical concept(s) that appear to be foundational to this problem.

Question
Math Word Problem
Find the greatest common divisor (a, b) and integers m and n such that (a, b) = am + bn.
a = 65, b = -91
b.) Explain the meaning of any notation used in the problem and in your solution.
c.) Describe the mathematical concept(s) that appear to be foundational to this problem.

2021-02-07
Step 1
a) The objective of the question is to find the gcd of two numbers and also to represent the gcd as a linear combination of the two numbers
b) the mathematical concept that will be used here is the Euclid's algorithm. It states that, given integers a,b,c find all integers x,y such that
c=xa+yb
Let d = gcd(a,b), and let b = b'd,a=a'd.Since xa+yb is a multiple of d for any integers x,y,solutions exist only when d divides c.
c) Given numbers are a = 65, b = -91.
Expressing the numbers as their prime factors,
a = 65 = 5 * 13
b = -91 = -7 * 13
Clearly, G.C.D.(65, -91) = 13.
Now, we want to find integers m and n such that 13 = 65m - 91n.
We shall proceed with Euclid's Algorithm. $$-{91}={65}{\left(-{2}\right)}+{\left({39}\right)},{0}\le{39}<{65}$$</span>
$$\Rightarrow{65}={39}{\left({1}\right)}+{26},{0}\le{26}<{39}$$</span>
$$\Rightarrow{26}{\left({1}\right)}+{13},{0}\le{13}<{26}$$</span>
$$\Rightarrow{26}={13}{\left({2}\right)}+{0}\le{0}<{13}$$</span>
Step 3
We have already established that the G.C.D.(65,-91) is 13.
Now, tracing back the steps we get,
13=39-26,
$$\Rightarrow{13}={39}-{\left({65}-{39}\right)}$$
$$\Rightarrow{13}={39}{\left({2}\right)}-{65}$$
$$\Rightarrow{13}={\left(-{91}+{65}{\left({2}\right)}\right)}{\left({2}\right)}-{65}$$
$$\Rightarrow{13}=-{91}{\left({2}\right)}+{65}{\left({3}\right)}$$
Thus the integers m and n are 3 and 2.

### Relevant Questions

b.)Explain the meaning of any notation used in the problem and in your solution.
c.)Describe the mathematical concept(s) that appear to be foundational to this problem.
d.)Justified solution to or proof of the problem.
Find the greatest common divisior of a,b, and c and write it in the form ax+by+cz for integers x,y, and z.
a=26,b=52,c=60
Worded problem: Follow these guided instructions to solve the worded problem below.
a) Assign a variable (name your variable)
b) write expression/s using your assigned variable,
d) Solve the algebraic inequality
Worded Inequality problem:
Your math test scores are 68, 78, 90 and 91. What is the lowest score you can earn on the next test and still achieve an average of at least 85?
To determine whether caffeine consumption affects the ability to solve math problems, a researcher had one group solve math problems after taking a cup of caffeinated drink and another group solve math problems after taking a cup of water. The group who took the caffeinated drink completed 35 problems in one hour and the group that had water completed 20 problems in one hour. Assuming the number of problems solved is normally distributed in each group, what statistical test would be used to test the research hypothesis? Explain your answer.
Explain : In your own words, describe a step-by-step approach for solving algebraic word problems.
Explain : In your own words, describe a step-by-step approach for solving algebraic word problems.
Montarello and Martins (2005) found that fifth grade students completed more mathematics problems correctly when simple problems were mixed in with their regular math assignments. To further explore this phenomenon, suppose that a researcher selects a standardized mathematics achievement test that produces a normal distribution of scores with a mean of $$\mu= 100\ and\ a\ standard\ deviation\ of\ \sigma = 24$$. The researcher modifies the test by inserting a set of very easy problems among the standardized questions and gives the modified test to a sample of n = 36 students. If the average test score for the sample is M = 120, is this result sufficient to conclude that inserting the easy questions improves student performance? Use a one-tailed test with $$\alpha = .05$$.
The null hypothesis in words is ?
Part II
29.[Poles] (a) For each of the pole diagrams below:
(i) Describe common features of all functions f(t) whose Laplace transforms have the given pole diagram.
(ii) Write down two examples of such f(t) and F(s).
The diagrams are: $$(1) {1,i,-i}. (2) {-1+4i,-1-4i}. (3) {-1}. (4)$$ The empty diagram.
(b) A mechanical system is discovered during an archaeological dig in Ethiopia. Rather than break it open, the investigators subjected it to a unit impulse. It was found that the motion of the system in response to the unit impulse is given by $$w(t) = u(t)e^{-\frac{t}{2}} \sin(\frac{3t}{2})$$
(i) What is the characteristic polynomial of the system? What is the transfer function W(s)?
(ii) Sketch the pole diagram of the system.
(ii) The team wants to transport this artifact to a museum. They know that vibrations from the truck that moves it result in vibrations of the system. They hope to avoid circular frequencies to which the system response has the greatest amplitude. What frequency should they avoid?
For Questions 1 — 2, use the following. Scooters are often used in European and Asian cities because of their ability to negotiate crowded city streets. The number of scooters (in thousands) sold each year in India can be approximated by the function $$N = 61.86t^2 — 237.43t + 943.51$$ where f is the number of years since 1990. 1. Find the vertical intercept. What is the practical meaning of the vertical intercept in this situation? 2. Use a numerical method to find the year when the number of scooters sold reaches 1 million. (Note that 1 million is 1,000 thousand, so N = 1000) Show three rows of the table you used to support your answer and write a clear answer to the problem.