By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. mixture of sand and iron, 20% is iron. 2^{2^6} &\equiv 16 \pmod{91} \\ Probability of Randomly Choosing a Prime Number - ThoughtCo Is the God of a monotheism necessarily omnipotent? Furthermore, all even perfect numbers have this form. Later entries are extremely long, so only the first and last 6 digits of each number are shown. This delves into complex analysis, in which there are graphs with four dimensions, where the fourth dimension is represented by the darkness of the color of the 3-D graph at its separate values. Then \(\frac{M_p\big(M_p+1\big)}{2}\) is an even perfect number. As new research comes out the answer to your question becomes more interesting. Or, is there some $n$ such that no primes of $n$-digits exist? If \(n\) is a prime number, then this gives Fermat's little theorem. numbers that are prime. Direct link to Sonata's post All numbers are divisible, Posted 12 years ago. By using our site, you rev2023.3.3.43278. What is the speed of the second train? I'll circle them. From 1 through 10, there are 4 primes: 2, 3, 5, and 7. How to deal with users padding their answers with custom signatures? For example, 2, 3, 5, 13 and 89. How far is the list of known primes known to be complete? \end{align}\]. I hope mod won't waste too much time on this. There are other issues, but this is probably the most well known issue. In a recent paper "Imperfect Forward Secrecy:How Diffie-Hellman Fails in Practice" by David Adrian et all found @ https://weakdh.org/imperfect-forward-secrecy-ccs15.pdf accessed on 10/16/2015 the researchers show that although there probably are a sufficient number of prime numbers available to RSA's 1024 bit key set there are groups of keys inside the whole set that are more likely to be used because of implementation. With a salary range between Rs. 3, so essentially the counting numbers starting How is the time complexity of Sieve of Eratosthenes is n*log(log(n))? This is due to the EuclidEuler theorem, partially proved by Euclid and completed by Leonhard Euler: even numbers are perfect if and only if they can be expressed in the form 2p 1 (2p 1), where 2p 1 is a Mersenne prime. * instead. But as you progress through There are $308,457,624,821$ 13 digit primes and $26,639,628,671,867$ 15 digit primes. Direct link to Jaguar37Studios's post It means that something i. Ltd.: All rights reserved, that can be divided exactly only by itself(example - 2, 3, 5, 7, 11 etc.). it in a different color, since I already used Learn more about Stack Overflow the company, and our products. So it won't be prime. How is an ETF fee calculated in a trade that ends in less than a year. For any integer \(n>3,\) there always exists at least one prime number \(p\) such that, This implies that for the \(k^\text{th}\) prime number, \(p_k,\) the next consecutive prime number is subject to. All you can say is that To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Is there a formula for the nth Prime? \[\begin{align} Direct link to Cameron's post In the 19th century some , Posted 10 years ago. The number of different orders in which books A, B and E may be arranged is, A school committee consists of 2 teachers and 4 students. one, then you are prime. The properties of prime numbers can show up in miscellaneous proofs in number theory. our constraint. And so it does not have Numbers that have more than two factors are called composite numbers. What will be the number of permutations of n different things, taken r at a time, where repeatition is allowed? Other examples of Fibonacci primes are 233 and 1597. Considering the answers it has already received it should've been closed as off-topic at security.SE and re-asked anew here. Palindromic number - Wikipedia Not 4 or 5, but it And maybe some of the encryption &= 2^4 \times 3^2 \\ Northern Coalfields Limited Fitter Mock Test, HAL Electronics - Management Trainees & Design Trainees Mock Test, FSSAI Technical Officer & Central Food Safety Officer Mock Test, DFCCIL Mechanical (Fitter) - Junior Executive Mock Test, IGCAR Mechanical - Technical Officer Mock Test, NMDC Maintenance Assistant Fitter Mock Test, IGCAR/NFC Electrician Stipendiary Trainee, BIS Mock Mock Test(Senior Secretariat Assistant & ASO), NIELIT (NIC) Technical Assistant Mock Test, Northern Coalfields Limited Previous Year Papers, FSSAI Technical Officer Previous Year Papers, AAI Junior Executive Previous Year Papers, DFCCIL Junior Executive Previous Year Papers, AAI JE Airport Operations Previous Year Papers, Vizag Steel Management Trainee Previous Year Papers, BHEL Engineer Trainee Previous Year Papers, NLC Graduate Executive Trainee Previous Year Papers, NPCIL Stipendiary Trainee Previous Year Papers, DFCCIL Junior Manager Previous Year Papers, NIC Technical Assistant A Previous Year Papers, HPCL Rajasthan Refinery Engineer Previous Year Papers, NFL Junior Engineering Assistant Grade II Previous Year Papers. In some sense, $2\%$ is small, but since there are $9\cdot 10^{21}$ numbers with $22$ digits, that means about $1.8\cdot 10^{20}$ of them are prime; not just three or four! What can a lawyer do if the client wants him to be acquitted of everything despite serious evidence? And hopefully we can In this video, I want Long division should be used to test larger prime numbers for divisibility. 119 is divisible by 7, so it is not a prime number. We've kind of broken How many prime numbers are there (available for RSA encryption)? Give the perfect number that corresponds to the Mersenne prime 31. Approach: The idea is to iterate through all the digits of the number and check whether the digit is a prime or not. In the following sequence, how many prime numbers are present? So in answer to your question there are probably a sufficient quantity of prime numbers in RSA encryption on paper but in practice there is a security issue if your hiding from a nation state. So 2 is prime. 39,100. To commemorate $50$ upvotes, here are some additional details: Bertrand's postulate has been proven, so what I've written here is not just conjecture. Is it suspicious or odd to stand by the gate of a GA airport watching the planes? This wouldn't be true if we considered 1 to be a prime number, because then someone else could say 24 = 3 x 2 x 2 x 2 x 1 and someone else could say 24 = 3 x 2 x 2 x 2 x 1 x 1 x 1 x 1 and so on, Sure, we could declare that 1 is a prime and then write an exception into the Fundamental Theorem of Arithmetic, but all in all it's less hassle to just say that 1 is neither prime nor composite. An emirp (prime spelled backwards) is a prime number that results in a different prime when its decimal digits are reversed. Direct link to Victor's post Why does a prime number h, Posted 10 years ago. Although one can keep going, there is seldom any benefit. And now I'll give All non-palindromic permutable primes are emirps. So, 6 is a perfect number because the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6. 233 is the only 3-digit Fibonacci prime and 1597 is also the case for the 4-digits. by exactly two natural numbers-- 1 and 5. These methods are called primality tests. The prime number theorem will give you a bound on the number of primes between $10^n$ and $10^{n+1}$. Not the answer you're looking for? Show that 7 is prime using Wilson's theorem. The next prime number is 10,007. And 16, you could have 2 times In contrast to prime numbers, a composite number is a positive integer greater than 1 that has more than two positive divisors. more in future videos. Prime numbers are numbers that have only 2 factors: 1 and themselves. Many theorems, such as Euler's theorem, require the prime factorization of a number. Why do small African island nations perform better than African continental nations, considering democracy and human development? Therefore, \(\phi(10)=4.\ _\square\). So, any combination of the number gives us sum of15 that will not be a prime number. about it-- if we don't think about the Is it impossible to publish a list of all the prime numbers in the range used by RSA? My C++ solution for Project Euler 35: Circular primes Sometimes, testing a number for primality does not involve exhaustively searching for prime factors, but instead making some clever observation about the number that leads to a factorization. A perfect number is a positive integer that is equal to the sum of its proper positive divisors. So 16 is not prime. First, let's find all combinations of five digits that multiply to 6!=720. Then. 94 is divided into two parts in such a way that the fifth part of the first and the eighth part of the second are in the ratio 3 : 4 The first part is: The denominator of a fraction is 4 more than twice the numerator. number factors. Compute \(a^{n-1} \bmod {n}.\) If the result is not \(1,\) then \(n\) is composite. I suggested to remove the unrelated comments in the question and some mod did it. This reduces the number of modular reductions by 4/5. [10], The following is a list of all currently known Mersenne primes and perfect numbers, along with their corresponding exponents p. As of 2022[update], there are 51 known Mersenne primes (and therefore perfect numbers), the largest 17 of which have been discovered by the distributed computing project Great Internet Mersenne Prime Search, or GIMPS. And if this doesn't The standard way to generate big prime numbers is to take a preselected random number of the desired length, apply a Fermat test (best with the base 2 as it can be optimized for speed) and then to apply a certain number of Miller-Rabin tests (depending on the length and the allowed error rate like 2100) to get a number which is very probably a Now, note that prime numbers between 1 and 10 are 2, 3, 5, 7. Show that 91 is composite using the Fermat primality test with the base \(a=2\). So if you can find anything pretty straightforward. How do you ensure that a red herring doesn't violate Chekhov's gun? The displayed ranks are among indices currently known as of 2022[update]; while unlikely, ranks may change if smaller ones are discovered. Prime numbers (video) | Khan Academy a little counter intuitive is not prime. Words are framed from the letters of the word GANESHPURI as follows, then the true statement is. If you don't know Can you write oxidation states with negative Roman numerals? 79. what people thought atoms were when see in this video, or you'll hopefully Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Allahabad University Group C Non-Teaching, Allahabad University Group B Non-Teaching, Allahabad University Group A Non-Teaching, NFL Junior Engineering Assistant Grade II, BPSC Asst. Which one of the following marks is not possible? Are there primes of every possible number of digits? this useful description of large prime generation, https://weakdh.org/imperfect-forward-secrecy-ccs15.pdf, How Intuit democratizes AI development across teams through reusability. 4 you can actually break with common difference 2, then the time taken by him to count all notes is. What is know about the gaps between primes? A chocolate box has 5 blue, 4 green, 2 yellow, 3 pink colored gems. Replacing broken pins/legs on a DIP IC package. Divide the chosen number 119 by each of these four numbers. How many circular primes are there below one million? say two other, I should say two The sum of the two largest two-digit prime numbers is \(97+89=186.\) \(_\square\). &= 2^2 \times 3^1 \\ \(_\square\), Let's work backward for \(n\). to talk a little bit about what it means My program took only 17 seconds to generate the 10 files. because one of the numbers is itself. My program took only 17 seconds to generate the 10 files. Connect and share knowledge within a single location that is structured and easy to search. I don't know whether it was due to math-phobia or due to something else but many important mathematically-oriented security-biased questions came to Math.SO (they should belong to Security.SO), a rabbit-rabbit problem at the best. That means that among these 10^150 numbers, there are approximately 10^150/ln(10^150) primes, which works out to 2.8x10^147 primes to choose from- certainly more than you could fit into any list!! \(_\square\). 7 is equal to 1 times 7, and in that case, you really For example, the prime gap between 13 and 17 is 4. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. Let \(p\) be prime. While the answer using Bertrand's postulate is correct, it may be misleading. Let's try out 3. Determine the fraction. Can you write oxidation states with negative Roman numerals? Testing primes with this theorem is very inefficient, perhaps even more so than testing prime divisors. The prime factorization of a positive integer is that number expressed as a product of powers of prime numbers. The question is still awfully phrased. So hopefully that And it's really not divisible These kinds of tests are designed to either confirm that the number is composite, or to use probability to designate a number as a probable prime. Books C and D are to be arranged first and second starting from the right of the shelf. 2^{2^3} &\equiv 74 \pmod{91} \\ It's not exactly divisible by 4. We conclude that moving to stronger key exchange methods should (You might ask why, in that case, we're not using this approach when we try and find larger and larger primes. The product of the digits of a five digit number is 6! I hope we can continue to investigate deeper the mathematical issue related to this topic. divisible by 5, obviously. But the, "which means the prime numbers range from 512 to 2048" - I think you mean 512 to 2048. UPSC NDA (I) Application Dates extended till 12th January 2023 till 6:00 pm. Why are Suriname, Belize, and Guinea-Bissau classified as "Small Island Developing States"? In the 19th century some mathematicians did consider 1 to be prime, but mathemeticians have found that it causes many problems in mathematics, if you consider 1 to be prime. Euler's totient function is critical for Euler's theorem. 97 is not divisible by 2, 3, 5, or 7, implying it is the largest two-digit prime number; 89 is not divisible by 2, 3, 5, or 7, implying it is the second largest two-digit prime number. Prime gaps tend to be much smaller, proportional to the primes. Another notable property of Mersenne primes is that they are related to the set of perfect numbers. One of the flags actually asked for deletion. This means that each positive integer has a prime factorization that no other positive integer has, and the order of factors in a prime factorization does not matter. Browse other questions tagged, Where developers & technologists share private knowledge with coworkers, Reach developers & technologists worldwide. (factorial). 1 is the only positive integer that is neither prime nor composite. How many more words (not necessarily meaningful) can be formed using the letters of the word RYTHM taking all at a time? Thumbs up :). Since it only guarantees one prime between $N$ and $2N$, you might expect only three or four primes with a particular number of digits. How to notate a grace note at the start of a bar with lilypond? Thanks for contributing an answer to Stack Overflow! This question is answered in the theorem below.) Frequently asked questions about primes - PrimePages The goal is to compute \(2^{90}\bmod{91}.\). There are other "traces" in a number that can indicate whether the number is prime or not. What can a lawyer do if the client wants him to be acquitted of everything despite serious evidence? 5 & 2^5-1= & 31 \\ Prime number: Prime number are those which are divisible by itself and 1. The last result that came out of GIMPS was $2^{74\,207\,281} - 1$, with over twenty million digits. Why are "large prime numbers" used in RSA/encryption? What are the prime numbers between 1 and 10? - Reviews Wiki | Source #1 Then, the user Fixee noticed my intention and suggested me to rephrase the question. not 3, not 4, not 5, not 6. 31. How many 4 digits numbers can be formed with the numbers 1, 3, 4, 5 ? of them, if you're only divisible by yourself and Posted 12 years ago. If a two-digit number is composite, then it must be divisible by a prime number that is less than or equal to \(\sqrt{100}=10.\) Therefore, it is sufficient to test 2, 3, 5, and 7 for divisibility. Use the method of repeated squares. it down as 2 times 2. \phi(2^4) &= 2^4-2^3=8 \\ Answer (1 of 5): [code]I think it is 99991 [/code]I wrote a sieve in python: [code]p = [True]*1000005 for x in range(2,40000): for y in range(x*2,1000001,x): p[y]=False [/code]Then searched the array for the last few primes below 100000 [code]>>> [x for x in range(99950,100000) if p. The next couple of examples demonstrate this. The number of different committees that can be formed from 5 teachers and 10 students is, If each element of a determinant of third order with value A is multiplied by 3, then the value of newly formed determinant is, If the coefficients of x7 and x8 in \(\left(2+\frac{x}{3}\right)^n\) are equal, then n is, The number of terms in the expansion of (x + y + z)10 is, If 2, 3 be the roots of 2x3+ mx2- 13x + n = 0 then the values of m and n are respectively, A person is to count 4500 currency notes. that your computer uses right now could be How many prime numbers are there (available for RSA encryption)? Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. be a little confusing, but when we see In Math.SO, Ross Millikan found the right words for the problem: semi-primes. (In fact, there are exactly $180,340,017,203,297,174,362$ primes with $22$ digits.). To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Direct link to Guy Edwards's post If you want an actual equ, Posted 12 years ago. whose first term is 2 and common difference 4, will be, The distance between the point P (2m, 3m, 4 m)and the x-axis is. divisible by 3 and 17. Count of Prime digits in a Number - GeeksforGeeks and the other one is one. if 51 is a prime number. Let's keep going, [1][5][6], It is currently an open problem as to whether there are an infinite number of Mersenne primes and even perfect numbers. natural numbers-- 1, 2, and 4. examples here, and let's figure out if some The number 1 is neither prime nor composite. m-hikari.com/ijcms-password/ijcms-password13-16-2006/, We've added a "Necessary cookies only" option to the cookie consent popup, Extending prime numbers digit by digit while retaining primality. It's not divisible by 2, so For instance, in the case of p = 2, 22 1 = 3 is prime, and 22 1 (22 1) = 2 3 = 6 is perfect. Previous . You can read them now in the comments between Fixee and me. Solution 1. . Prime numbers are critical for the study of number theory. This question appears to be off-topic because it is not about programming. If our prime has 4 or more digits, and has 2 or more not equal to 3, we can by deleting one or two get a number greater than 3 with digit sum divisible by 3. How many three digit palindrome number are prime? to be a prime number. Thus, there is a total of four factors: 1, 3, 5, and 15. 3 digit Prime Palindrome Numbers. - Mathematics Stack Exchange There are only finitely many, indeed there are none with more than 3 digits. Why does a prime number have to be divisible by two natural numbers? Clearly our prime cannot have 0 as a digit. the second and fourth digit of the number) . 2^{90} &\equiv (16)(16)(74)(4) \pmod{91} \\ For example, 5 is a prime number because it has no positive divisors other than 1 and 5. \text{lcm}(36,48) &= 2^{\max(2,4)} \times 3^{\max(2,1)} \\ Any 3 digit palindrome number is of type "aba" where b can be chosen from the numbers 0 to 9 and a can be chosen from 1 to 9. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. [2][4], There is a one-to-one correspondence between the Mersenne primes and the even perfect numbers. For instance, I might say that 24 = 3 x 2 x 2 x 2 and you might say 24 = 2 x 2 x 3 x 2, but we each came up with three 2's and one 3 and nobody else could do differently. . I need a few small primes (say 10 to 300 digits) Mersenne Numbers What are the known Mersenne primes? How to Create a List of Primes Using the Sieve of Eratosthenes All numbers are divisible by decimals. A second student scores 32% marks but gets 42 marks more than the minimum passing marks. 1. get the right-most digit: auto digit = rotated % 10; 2. move all digits by one digit to the right ("erasing" the right-most digit): rotated /= 10; 3. prepend the right-most digit: rotated += digit * shift; 4. check whether rotated is part of our std::set, too 5. if rotated is equal to our initial value x then we checked all rotations A 5 digit number using 1, 2, 3, 4 and 5 without repetition. For any real number \(x,\) \(\pi(x)\) gives the number of prime numbers that are less than or equal to \(x.\) Then, \[\lim_{x \rightarrow \infty} \frac{\hspace{2mm} \pi(x)\hspace{2mm} }{\frac{x}{\ln{x}}}=1.\], This implies that for sufficiently large \(x,\). Prime factorization is the primary motivation for studying prime numbers. Direct link to eleanorwong135's post Why is 2 considered a pri, Posted 10 years ago. From 11 through 20, there are again 4 primes: 11, 13, 17, and 19. The RSA method of encryption relies upon the factorization of a number into primes. Explanation: Digits of the number - {1, 2} But, only 2 is prime number. It only takes a minute to sign up. Direct link to Matthew Daly's post The Fundamental Theorem o, Posted 11 years ago. It's also divisible by 2. For every prime number p, there exists a prime number p' such that p' is greater than p. This mathematical proof, which was demonstrated in ancient times by the . agencys attacks on VPNs are consistent with having achieved such a You can't break And if you're I guess you could Prime and Composite Numbers Prime Numbers - Advanced Thus, any prime \(p > 3\) can be represented in the form \(6k+5\) or \(6k+1\). This is the complete index for the prime curiosity collection--an exciting collection of curiosities, wonders and trivia related to prime numbers and integer factorization. 3 is also a prime number. It is a natural number divisible rev2023.3.3.43278. However, this theorem does give insight that a number's primality is not linked purely to the divisors of that number. So, 15 is not a prime number. If a, b, c, d are in H.P., then the value of\(\left(\frac{1}{a^2}-\frac{1}{d^2}\right)\left(\frac{1}{b^2}-\frac{1}{c^2}\right) ^{-1} \)is: The sum of 40 terms of an A.P. Direct link to Fiona's post yes. However, I was thinking that result would make total sense if there is an $n$ such that there are no $n$-digit primes, since any $k$-digit truncatable prime implies the existence of at least one $n$-digit prime for every $n\leq k$. Where does this (supposedly) Gibson quote come from? 1 and 17 will Candidates who get successful selection under UPSC NDA will get a salary range between Rs. m&=p_1^{j_1} \times p_2^{j_2} \times p_3^{j_3} \times \cdots\\ In how many different ways this canbe done? Each repetition of these steps improves the probability that the number is prime. &\vdots\\ \end{align}\]. Prime numbers act as "building blocks" of numbers, and as such, it is important to understand prime numbers to understand how numbers are related to each other. Wouldn't there be "commonly used" prime numbers? With the side note that Bertrand's postulate is a (proved) theorem. 840. The vale of the expresssion\(\frac{2.25^2-1.25^2}{2.25-1.25}\)is. haven't broken it down much. Yes, there is always such a prime. Forgot password? However, if \(q\) and \(r\) are both greater than \(\sqrt{n},\) then \(qr>n.\) This cannot be true, because \(n=kqr,\) and \(k\) is a positive integer. You might be tempted 5 Digit Prime Numbers List - PrimeNumbersList.com number you put up here is going to be The Dedicated Freight Corridor Corporation of India Limited (DFCCIL) has released the DFCCIL Junior Executive Result for Mechanical and Signal & Telecommunication against Advt No. break. again, just as an example, these are like the numbers 1, 2, The highest power of 2 that 48 is divisible by is \(16=2^4.\) The highest power of 3 that 48 is divisible by is \(3=3^1.\) Thus, the prime factorization of 48 is, The fundamental theorem of arithmetic guarantees that no other positive integer has this prime factorization. \phi(48) &= 8 \times 2=16.\ _\square How do you get out of a corner when plotting yourself into a corner. Bertrand's postulate states that for any $k>3$, there is a prime between $k$ and $2k-2$. Anyway, yes: for all $n$ there are a lot of primes having $n$ digits. So 17 is prime. Thus the probability that a prime is selected at random is 15/50 = 30%. is divisible by 6. Now with that out of the way, 97. The total number of 3-digit numbers that can be formed = 555 = 125. So let's start with the smallest another color here. You could divide them into it, 1234321&= 11111111\\ 2^{90} &= 2^{2^6} \times 2^{2^4} \times 2^{2^3} \times 2^{2^1} \\\\ Finally, prime numbers have applications in essentially all areas of mathematics. You might say, hey, video here and try to figure out for yourself Prime factorizations can be used to compute GCD and LCM. One thing that annoys me is that the non-math-answers penetrated to Math.SO with high-scores, distracting the discussion. Five different books (A, B, C, D and E) are to be arranged on a shelf. If \(n\) is a power of a prime, then Euler's totient function can be computed efficiently using the following theorem: For any given prime \(p\) and positive integer \(n\). So I'll give you a definition. 211 is not divisible by any of those numbers, so it must be prime. According to GIMPS, all possibilities less than the 48th working exponent p = 57,885,161 have been checked and verified as of October2021[update]. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 71. Are there number systems or rings in which not every number is a product of primes? That means that your prime numbers are on the order of 2^512: over 150 digits long. Bulk update symbol size units from mm to map units in rule-based symbology. Very good answer. Sanitary and Waste Mgmt. I'm confused. Fortunately, one does not need to test the divisibility of each smaller prime to conclude that a number is prime. That question mentioned security, trust, asked whether somebody could use the weakness to their benefit, and how to notify the bank of a problem. This should give you some indication as to why . How to handle a hobby that makes income in US. yes. 121&= 1111\\ be a priority for the Internet community. Let \(\pi(x)\) be the prime counting function. [3] Meanwhile, perfect numbers are natural numbers that equal the sum of their positive proper divisors, which are divisors excluding the number itself. Is it possible to create a concave light? How to tell which packages are held back due to phased updates. By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. Direct link to Peter Collingridge's post Neither - those terms onl, Posted 10 years ago. Just another note: those interested in this sort of thing should look for papers by Pierre Dusart - he has proven many of the best approximations of this form. to think it's prime. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. thing that you couldn't divide anymore. natural number-- the number 1. Since the only divisors of \(p\) are \(1\) and \(p,\) and \(p\) doesn't divide \(a,\) we must have \(\gcd (a, p) =1.\) By Bezout's identity, there exist some \(u\) and \(v\) such that \(ua+vp=1\).
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