# Prove the Fundamental Theorem of Arithmetic. Every integer than 1 is a prime or a product of primes. This product is unique, exept for the order in which the factors appear.

Question
Polynomial arithmetic
Prove the Fundamental Theorem of Arithmetic. Every integer than 1 is a prime or a product of primes. This product is unique, exept for the order in which the factors appear.

2021-02-26
The objective is to prove the Fundamental Theorem of Arithmetic, that states: “Every number greater than 1 is a prime or a product of primes. This product is unique, except for the order in which the factors appear.” To prove the existence, use induction: 2>1 is a prime number itself so it satisfy the statement. Now let all the integers from 1 to k, i.e. 1$$\displaystyle{k}={a}{b}:$$ where $$\displaystyle{1}{<}{a},{\quad\text{and}\quad}\ {b}{<}{k}$$</span> By our asumption since both a and b are between 1 and k they both can be written as product of primes So, k = ab can be written as product of primes. Thus, by induction all integers gratee than 1 are either prime or can be written as product of primes. Now, let’s prove uniqueness of the product for that let a integer n can be expressed as a product of primes in two ways: $$\displaystyle{n}={p}_{{1}}\cdot{p}_{{2}}\cdot\ldots.\cdot{p}_{{k}},{\quad\text{and}\quad}{n}={s}_{{1}}\cdot{s}_{{2}}\cdot\ldots\cdot{s}+{m}$$
Where, $$\displaystyle{p}_{{1}},{p}_{{2}},\ldots.,{p}_{{k}}{\quad\text{and}\quad}{s}_{{1}},{s}_{{2}},\ldots,{s}_{{m}}$$ are primes.
$$\displaystyle\Rightarrow{n}={p}_{{1}}\cdot{p}_{{2}}\cdot\ldots.\cdot{p}_{{k}}={s}_{{1}}\cdot{s}_{{2}}\cdot\ldots\cdot{s}_{{m}}\Rightarrow{p}_{{i}}{\mid}{s}_{{1}}\cdot{s}_{{2}}\cdot\ldots\cdot{s}_{{m}}{f}{\quad\text{or}\quad}{a}{n}{y}{1}\leq{i}{<}{k}$$</span>
Now, by Euclid’s Lemma $$\displaystyle\exists{s}_{{j}}$$ for $$\displaystyle{1}\leq{j}{<}{m}$$</span> such that $$\displaystyle{p}_{{i}}{\mid}{s}_{{j}}$$ And since both $$\displaystyle{p}_{{i}},$$ and $$\displaystyle{s}_{{j}}$$ are primes $$\displaystyle\Rightarrow{p}_{{i}}={s}_{{j}}$$ Divide these two common factors from $$\displaystyle{p}_{{1}}\cdot{p}_{{2}}\cdot\ldots\cdot{p}_{{k}}={s}_{{1}}\cdot{s}_{{2}}\cdot\ldots\cdot{s}_{{m}}$$ and repeat the process untill all common factors have been divided out. Also note that k = m unless then after repeating above process we will be left with $$\displaystyle{p}_{{{i}_{{1}}}}\cdot{p}_{{{i}_{{2}}}}\cdot\ldots\cdot{p}_{{{i}_{{r}}}}={1}{\left({\quad\text{if}\quad}{k}{>}{m}\right)}{\quad\text{or}\quad}{s}_{{{m}_{{1}}}}\cdot{s}_{{{m}_{{2}}}}\cdot\ldots\cdot{s}_{{{m}_{{n}}}}={1}{\left({\quad\text{if}\quad}{k}{<}{m}\right)}$$</span> And, all primes are greater than 1 so both of above cases are not possible, thus, $$\displaystyle{k}={m}.$$ And after repeating the process we have $$\displaystyle{p}_{{i}}={s}_{{r}}\forall{1}\leq{i}\leq{k},{\quad\text{and}\quad}{1}\leq{r}\leq{m}$$ Thus, product of prime factors of a number is unique.

### Relevant Questions

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a. List a rearranged order of the five traditional stages of the RISC pipeline that will support register-memory operations implemented exclusively by register indirect addressing.
b. Describe what new forwarding paths are needed for the rearranged pipeline by stating the source, destination, and information transferred on each needed new path.
c. For the reordered stages of the RISC pipeline, what new data hazards are created by this addressing mode? Give an instruction sequence illustrating each new hazard.
d. List all of the ways that the RISC pipeline with register-memory ALU operations can have a different instruction count for a given program than the original RISC pipeline. Give a pair of specific instruction sequences, one for the original pipeline and one for the rearranged pipeline, to illustrate each way.
Hint for (d): Give a pair of instruction sequences where the RISC pipeline has “more” instructions than the reg-mem architecture. Also give a pair of instruction sequences where the RISC pipeline has “fewer” instructions than the reg-mem architecture.
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$$Q(x) = (((3x - 5)x + 1)x 3)x + 5$$
Try to evaluate P(2) and Q(2) in your head, using the forms given. Which is easier? Now write the polynomial
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Do you see how calculating with the nested form follows the same arithmetic steps as calculating the value ofa polynomial using synthetic division?
State the fundamental theorem of arithmetic.

A random sample of $$n_1 = 14$$ winter days in Denver gave a sample mean pollution index $$x_1 = 43$$.
Previous studies show that $$\sigma_1 = 19$$.
For Englewood (a suburb of Denver), a random sample of $$n_2 = 12$$ winter days gave a sample mean pollution index of $$x_2 = 37$$.
Previous studies show that $$\sigma_2 = 13$$.
Assume the pollution index is normally distributed in both Englewood and Denver.
(a) State the null and alternate hypotheses.
$$H_0:\mu_1=\mu_2.\mu_1>\mu_2$$
$$H_0:\mu_1<\mu_2.\mu_1=\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1<\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1\neq\mu_2$$
(b) What sampling distribution will you use? What assumptions are you making? NKS The Student's t. We assume that both population distributions are approximately normal with known standard deviations.
The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.
(c) What is the value of the sample test statistic? Compute the corresponding z or t value as appropriate.
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(e) Based on your answers in parts (i)−(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \alpha?
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are not statistically significant.
(f) Interpret your conclusion in the context of the application.
Reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver. (g) Find a 99% confidence interval for
$$\mu_1 - \mu_2$$.
lower limit
upper limit
(h) Explain the meaning of the confidence interval in the context of the problem.
Because the interval contains only positive numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, we can not say that the mean population pollution index for Englewood is different than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains only negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is less than that of Denver.
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