# 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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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.
$$H_0:\mu_1=\mu_2.\mu_1>\mu_2$$
$$H_0:\mu_1<\mu_2.\mu_1=\mu_2$$
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$$H_0:\mu_1=\mu_2.\mu_1\neq\mu_2$$
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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.
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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
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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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