A mathematician is trying to break down a large number into smaller prime numbers that, when multiplied, would arrive at that larger number. what is this process called?

Prime factorization is a method of breaking large numbers down into smaller prime numbers that when multiplied together equal the original number. Any non-prime number can be broken down into its prime factors, but breaking down these numbers is hard! This tool will make it easier and help you check your work when you need to do it by hand. To break down a number (like 462) into its prime factors by hand you would: • First check to see if 462 is divisible by the smallest prime number over 1, which is 2. • If your number is divisible by 2, you would write 2 and 462/2 or 231 down. • Then you would take the 231 and try to divide by 2 again. • Since 231 is not divisible by 2, you would then check the next highest prime number, which is 3. 231 is divisible by 3 (remember your • 77 is not a prime number, so we need to keep going. You will take 77 and try to divide by prime numbers again. Both 2 and 3, plus 5, the next prime number, do not divide evenly into 77, but 7, the next prime number after 5, does divide evenly. 77/7 equals 11 so you would write 7 and 11 down. • When we look at 11, since it is a prime number, we’re done! The prime factorization of 462 is thus 2,3,7,11. • Try putting our original number into this calculator and you’ll see that we get the same result. Huzzah! welcome to coolmath We use first party cookies on our website to enhance your browsing experience, and third party cookies to provide advertising that may be of interest to you. You can accept or reject...

On April 17, a paper arrived in the inbox of Annals of Mathematics, one of the discipline’s preeminent journals. Written by a mathematician virtually unknown to the experts in his field — a 50-something lecturer at the University of New Hampshire named Yitang Zhang — the paper claimed to have taken a huge step forward in understanding one of mathematics’ oldest problems, the twin primes conjecture. Simons Science News , an editorially independent division of Just three weeks later — a blink of an eye compared to the usual pace of mathematics journals — Zhang received the referee report on his paper. Yitang Zhang (Photo: University of New Hampshire) Lisa Nugent Rumors swept through the mathematics community that a great advance had been made by a researcher no one seemed to know — someone whose talents had been so overlooked after he earned his doctorate in 1991 that he had found it difficult to get an academic job, working for several years as an accountant and even in a Subway sandwich shop. Mathematicians at Harvard University hastily arranged for Zhang to present his work to a packed audience there on May 13. As details of his work have emerged, it has become clear that Zhang achieved his result not via a radically new approach to the problem, but by applying existing methods with great perseverance. “The big experts in the field had already tried to make this approach work,” Granville said. “He’s not a known expert, but he succeeded where all the experts had failed.” T...

This problem has been solved: Solutions for Chapter 9 Problem 16ST: A mathematician is trying to figure out if there is a relationship between the number of vertices, n, in a graph and the product of the graph’s independence and clique numbers. (We knowfromExercise 52.8 that , so perhaps we can show that ω(G) ≥ n/α(G).) Alas, there is no good relation. Demonstrate this by finding three graphs with the following properties: Extra kudos if your three examples all have the same number of vertices. … Get solutions Get solutions Get solutions done loading • CH1 • CH1.1 • CH1.2 • CH1.3 • CH1.4 • CH1.5 • CH1.6 • CH1.7 • CH2 • CH2.8 • CH2.9 • CH2.10 • CH2.11 • CH2.12 • CH2.13 • CH3 • CH3.14 • CH3.15 • CH3.16 • CH3.17 • CH3.18 • CH3.19 • CH4 • CH4.20 • CH4.21 • CH4.22 • CH4.23 • CH5 • CH5.24 • CH5.25 • CH5.26 • CH5.27 • CH5.28 • CH5.29 • CH6 • CH6.30 • CH6.31 • CH6.32 • CH6.33 • CH6.34 • CH7 • CH7.35 • CH7.36 • CH7.37 • CH7.38 • CH7.39 • CH8 • CH8.40 • CH8.41 • CH8.42 • CH8.43 • CH8.44 • CH8.45 • CH8.46 • CH9 • CH9.47 • CH9.48 • CH9.49 • CH9.50 • CH9.51 • CH9.52 • CH9.53 • CH10 • CH10.54 • CH10.55 • CH10.56 • CH10.57 • CH10.58 • CH10.59 A mathematician is trying to figure out if there is a relationship between the number of vertices, n, in a graph and the product of the graph’s independence and clique numbers. (We knowfromExercise 52.8 that , so perhaps we can show that ω(G) ≥ n/α(G).) Alas, there is no good relation. Demonstrate this by finding three graphs with the following prop...

Prime numbers are whole numbers (1,2,3, and so on) bigger than 1, which have the property that they can’t be written as two smaller numbers multiplied together. So 6 isn’t prime since it can be written as 2×3, but 5 is prime since the only way of writing it as two whole numbers multiplied together is 1×5 or 5×1. Another way of thinking about this is if you have some coins, but you can’t arrange all of them into a rectangle other than by putting them in a straight line, then the number of coins is a prime number. An Infinite Number Of Primes It might sound strange why anyone would care about such numbers, but it turns out they’re fundamental to mathematics. This is because every number can be written in a unique way, as prime numbers are multiplied together. This means that prime numbers are the `atoms of multiplication’ – the smallest things which everything else can be built from. Because primes are the building blocks of whole numbers by multiplication, many problems about whole numbers can be reduced to a problem about prime numbers. In much the same way, some problems in Chemistry can be solved by understanding the atomic composition of the chemicals involved. Thus, if there were a finite number of primes, one could possibly then check each one in turn on a computer. However, it turns out that there are infinitely many prime numbers and that prime numbers are still surprisingly poorly understood by mathematicians. Mathematician Gilbert Strang on the difference between ...

When Daniel Larsen was in middle school, he started designing crossword puzzles. He had to layer the hobby on top of his other interests: chess, programming, piano, violin. He twice qualified for the Scripps National Spelling Bee near Washington, D.C., after winning his regional competition. “He gets focused on something, and it’s just bang, bang, bang, until he succeeds,” said Larsen’s mother, Ayelet Lindenstrauss. His first crossword puzzles were rejected by major newspapers, but he kept at it and ultimately broke in. To date, he The New York Times, at age 13. “He’s very persistent,” Lindenstrauss said. Still, Larsen’s most recent obsession felt different, “longer and more intense than most of his other projects,” she said. For more than a year and a half, Larsen couldn’t stop thinking about a certain math problem. It had roots in a broader question, one that the mathematician Carl Friedrich Gauss considered to be among the most important in mathematics: how to distinguish a prime number (a number that is divisible only by 1 and itself) from a composite number. For hundreds of years, mathematicians have sought an efficient way to do so. The problem has also become relevant in the context of modern cryptography, as some of today’s most widely used cryptosystems involve doing arithmetic with enormous primes. Over a century ago, in that quest for a fast, powerful primality test, mathematicians stumbled on a group of troublemakers — numbers that fool tests into thinking they...

More than 3,550 years ago, an Egyptian scribe named Ahmes The mathematic papyrus of Amhes. Credit: Paul James Cowie The definition of a prime number is so simple that it is learned in primary school: it is that natural number greater than 1 that can only be divided exactly by 1 and by itself. In fact, this apparent simplicity is part of its appeal, according to what Adrian Dudek, a mathematician from Australian National University, tells OpenMind: “I think the fascination for prime numbers comes from the fact that they are so elementary in description but yet incredibly difficult to analyse. A young child can understand what makes a number prime, yet lifetimes of mathematical research have been spent trying to solve some of the problems in the field.” The first known person to look specifically at this subject was the Greek mathematician Euclid of Alexandria, who around 300 B.C. demonstrated for the first time that prime numbers are infinite. A century later, another Greek mathematician, The Mersenne primes After the Greeks, interest in prime numbers was only revived at the end of the Middle Ages. At the beginning of the 17th century, French monk Marin Mersenne defined the prime numbers that bear his name, obtained as M p = 2 p – 1. If p is a prime number, it is possible, though not certain, that M p is also a prime number. Already in 1588, Italian mathematician Pietro Cataldi had shown that 2 19 – 1 = 524,287 is prime, setting a record for his time. 127 – 1 is a prime. Th...

When Daniel Larsen was in middle school, he started designing crossword puzzles. He had to layer the hobby on top of his other interests: chess, programming, piano, violin. He twice qualified for the Scripps National Spelling Bee near Washington, D.C., after winning his regional competition. “He gets focused on something, and it’s just bang, bang, bang, until he succeeds,” said Larsen’s mother, Ayelet Lindenstrauss. His first crossword puzzles were rejected by major newspapers, but he kept at it and ultimately broke in. To date, he The New York Times, at age 13. “He’s very persistent,” Lindenstrauss said. Still, Larsen’s most recent obsession felt different, “longer and more intense than most of his other projects,” she said. For more than a year and a half, Larsen couldn’t stop thinking about a certain math problem. It had roots in a broader question, one that the mathematician Carl Friedrich Gauss considered to be among the most important in mathematics: how to distinguish a prime number (a number that is divisible only by 1 and itself) from a composite number. For hundreds of years, mathematicians have sought an efficient way to do so. The problem has also become relevant in the context of modern cryptography, as some of today’s most widely used cryptosystems involve doing arithmetic with enormous primes. Over a century ago, in that quest for a fast, powerful primality test, mathematicians stumbled on a group of troublemakers — numbers that fool tests into thinking they...

This problem has been solved: Solutions for Chapter 9 Problem 16ST: A mathematician is trying to figure out if there is a relationship between the number of vertices, n, in a graph and the product of the graph’s independence and clique numbers. (We knowfromExercise 52.8 that , so perhaps we can show that ω(G) ≥ n/α(G).) Alas, there is no good relation. Demonstrate this by finding three graphs with the following properties: Extra kudos if your three examples all have the same number of vertices. … Get solutions Get solutions Get solutions done loading • CH1 • CH1.1 • CH1.2 • CH1.3 • CH1.4 • CH1.5 • CH1.6 • CH1.7 • CH2 • CH2.8 • CH2.9 • CH2.10 • CH2.11 • CH2.12 • CH2.13 • CH3 • CH3.14 • CH3.15 • CH3.16 • CH3.17 • CH3.18 • CH3.19 • CH4 • CH4.20 • CH4.21 • CH4.22 • CH4.23 • CH5 • CH5.24 • CH5.25 • CH5.26 • CH5.27 • CH5.28 • CH5.29 • CH6 • CH6.30 • CH6.31 • CH6.32 • CH6.33 • CH6.34 • CH7 • CH7.35 • CH7.36 • CH7.37 • CH7.38 • CH7.39 • CH8 • CH8.40 • CH8.41 • CH8.42 • CH8.43 • CH8.44 • CH8.45 • CH8.46 • CH9 • CH9.47 • CH9.48 • CH9.49 • CH9.50 • CH9.51 • CH9.52 • CH9.53 • CH10 • CH10.54 • CH10.55 • CH10.56 • CH10.57 • CH10.58 • CH10.59 A mathematician is trying to figure out if there is a relationship between the number of vertices, n, in a graph and the product of the graph’s independence and clique numbers. (We knowfromExercise 52.8 that , so perhaps we can show that ω(G) ≥ n/α(G).) Alas, there is no good relation. Demonstrate this by finding three graphs with the following prop...

Prime numbers are whole numbers (1,2,3, and so on) bigger than 1, which have the property that they can’t be written as two smaller numbers multiplied together. So 6 isn’t prime since it can be written as 2×3, but 5 is prime since the only way of writing it as two whole numbers multiplied together is 1×5 or 5×1. Another way of thinking about this is if you have some coins, but you can’t arrange all of them into a rectangle other than by putting them in a straight line, then the number of coins is a prime number. An Infinite Number Of Primes It might sound strange why anyone would care about such numbers, but it turns out they’re fundamental to mathematics. This is because every number can be written in a unique way, as prime numbers are multiplied together. This means that prime numbers are the `atoms of multiplication’ – the smallest things which everything else can be built from. Because primes are the building blocks of whole numbers by multiplication, many problems about whole numbers can be reduced to a problem about prime numbers. In much the same way, some problems in Chemistry can be solved by understanding the atomic composition of the chemicals involved. Thus, if there were a finite number of primes, one could possibly then check each one in turn on a computer. However, it turns out that there are infinitely many prime numbers and that prime numbers are still surprisingly poorly understood by mathematicians. Mathematician Gilbert Strang on the difference between ...

More than 3,550 years ago, an Egyptian scribe named Ahmes The mathematic papyrus of Amhes. Credit: Paul James Cowie The definition of a prime number is so simple that it is learned in primary school: it is that natural number greater than 1 that can only be divided exactly by 1 and by itself. In fact, this apparent simplicity is part of its appeal, according to what Adrian Dudek, a mathematician from Australian National University, tells OpenMind: “I think the fascination for prime numbers comes from the fact that they are so elementary in description but yet incredibly difficult to analyse. A young child can understand what makes a number prime, yet lifetimes of mathematical research have been spent trying to solve some of the problems in the field.” The first known person to look specifically at this subject was the Greek mathematician Euclid of Alexandria, who around 300 B.C. demonstrated for the first time that prime numbers are infinite. A century later, another Greek mathematician, The Mersenne primes After the Greeks, interest in prime numbers was only revived at the end of the Middle Ages. At the beginning of the 17th century, French monk Marin Mersenne defined the prime numbers that bear his name, obtained as M p = 2 p – 1. If p is a prime number, it is possible, though not certain, that M p is also a prime number. Already in 1588, Italian mathematician Pietro Cataldi had shown that 2 19 – 1 = 524,287 is prime, setting a record for his time. 127 – 1 is a prime. Th...