# Mathematical Operations With Prime Numbers

## Prime or composite numbers? The last 3 numbers on which the prime factorization has been performed

 What is the prime factorization of the composite number 2,457,959? Dec 03 23:34 UTC (GMT) What is the prime factorization of the composite number 51? Dec 03 23:34 UTC (GMT) What is the prime factorization of the composite number 37,636? Dec 03 23:34 UTC (GMT) The list of numbers that were checked on whether they are prime or not. The prime factorization operations of the composite numbers.

## The greatest (highest) common factor (divisor), gcf (hcf, gcd): the latest 3 calculated values

 What is the greatest (highest) common factor (divisor) of the numbers 7,674 and 3,248? How to calculate the GCF (HCF, GCD)? Dec 03 23:34 UTC (GMT) What is the greatest (highest) common factor (divisor) of the numbers 81 and 72? How to calculate the GCF (HCF, GCD)? Dec 03 23:34 UTC (GMT) What is the greatest (highest) common factor (divisor) of the numbers 26 and 91? How to calculate the GCF (HCF, GCD)? Dec 03 23:34 UTC (GMT) The greatest (highest) common factor (divisor), gcf (hcf, gcd): the list of all the calculations

## The least (the lowest) common multiple, LCM: the latest 3 calculated values

 What is the least (the lowest) common multiple, LCM, of the numbers 6,625 and 5,283 and how to calculate it? Dec 03 23:34 UTC (GMT) What is the least (the lowest) common multiple, LCM, of the numbers 88 and 46 and how to calculate it? Dec 03 23:34 UTC (GMT) What is the least (the lowest) common multiple, LCM, of the numbers 12 and 20 and how to calculate it? Dec 03 23:33 UTC (GMT) The least (the lowest) common multiple, LCM: the list of all the operations

## The latest 3 fractions that have been fully reduced (simplified) to their lowest terms (to their simplest form, the smallest possible numerator and denominator)

 Completely reduce (simplify) the fraction 6,370 / 2,275 to the lowest terms (to its simplest form, irreducible - the smallest possible numerator and denominator) Dec 03 23:34 UTC (GMT) Completely reduce (simplify) the fraction 20 / 200 to the lowest terms (to its simplest form, irreducible - the smallest possible numerator and denominator) Dec 03 23:34 UTC (GMT) Completely reduce (simplify) the fraction 4,028 / 864 to the lowest terms (to its simplest form, irreducible - the smallest possible numerator and denominator) Dec 03 23:34 UTC (GMT) The list of all the fractions that were fully reduced (simplified) to their lowest terms (to their simplest form), the smallest possible numerator and denominator

## Divisibility: the latest 3 pairs of numbers checked on whether they are divisible or not

 Is the number 18,444 divisible by 4? Could 18,444 be evenly divided by 4? Does the first number contain all the prime factors of the second? Dec 03 23:34 UTC (GMT) Is the number 7,199 divisible by 292? Could 7,199 be evenly divided by 292? Does the first number contain all the prime factors of the second? Dec 03 23:34 UTC (GMT) Is the number 9 divisible by 264? Could 9 be evenly divided by 264? Does the first number contain all the prime factors of the second? Dec 03 23:34 UTC (GMT) The list of all the pairs of numbers that were checked on whether they are divisible or not

## The latest 3 sets of calculated factors (divisors): of one number or the common factors of two numbers

 What are all the common factors (all the divisors and the prime factors) of the numbers 1,775,369 and 0? How to calculate them? Dec 03 23:34 UTC (GMT) What are all the common factors (all the divisors and the prime factors) of the numbers 5,352,050 and 0? How to calculate them? Dec 03 23:34 UTC (GMT) What are all the proper, improper and prime factors (all the divisors) of the number 5,925,880? How to calculate them? Dec 03 23:34 UTC (GMT) The list of all the calculated factors (divisors) of one or two numbers

## The latest 3 pairs of numbers checked on whether they are coprime (prime to each other, relatively prime) or not

 Are the two numbers 1,296 and 5,660 coprime (relatively prime, prime to each other) or not? Dec 03 23:33 UTC (GMT) Are the two numbers 3,665 and 8,141 coprime (relatively prime, prime to each other) or not? Dec 03 23:31 UTC (GMT) Are the two numbers 31,488,155,893 and 2,389 coprime (relatively prime, prime to each other) or not? Dec 03 23:31 UTC (GMT) All the pairs of numbers that were checked on whether they are coprime (prime to each other, relatively prime) or not

## The latest 3 operations on numbers' parity: even or odd numbers?

 Numbers parity: Is 43,578,070 an even or an odd number? Dec 03 23:33 UTC (GMT) Numbers parity: Is 3,500 an even or an odd number? Dec 03 23:33 UTC (GMT) Numbers parity: Is 610,814 an even or an odd number? Dec 03 23:33 UTC (GMT) The list of all the checked on numbers: is it an even or an odd number?

## 1. Prime numbers. 2. The fundamental theorem of arithmetic. 3. Composite numbers. 4. Remarks

• ### 1. Prime numbers

• A prime number is a natural number, larger than 1, which is evenly dividing (= without a remainder) only by 1 and itself.
• Any "m" prime number has only two divisors (two factors): the number itself, "m", and the number 1.
• Examples of prime numbers:
• 1 is not considered a prime number, so the first prime number is 2 (the prime numbers list is starting with the number 2).
• 2 is divisible only by 2 and 1, so 2 is a prime number.
• 3 is divisible only by 3 and 1, so 3 is a prime number.
• 5 is divisible only by 5 and 1, so 5 is a prime number.
• 13 is divisible only by 13 and 1, so 13 is a prime number.
• ### 2. The fundamental theorem of arithmetic

• The fundamental theorem of arithmetic says that every natural number larger than 1 can be written as a product of one or more prime numbers in a way that is unique, except for the order of the prime factors.
• Why is 1 not considered a prime number? If 1 were considered a prime number, then the prime factorization of the number 15, for example, could be either: 15 = 3 × 5 or 15 = 1 × 3 × 5. These two representations would have been considered two different prime factorizations of the same number, 15, so the statement of the fundamental theorem would no longer be true.
• ### 3. Composite numbers

• A composite number is a natural number that has at least one positive divisor (factor) other than 1 and the number itself.
• A composite number is also any number larger than 1 that is not a prime number.
• The Prime Factorization of a number: finding the prime numbers that multiply together to make that number.
• Examples of composite numbers:
• 4 is divisible by 4, 2 and 1, so 4 is not a prime number, it is a composite number. The prime factorization of 4 = 2 × 2 = 22
• First Note: The second part of the prime factorization of 4 is written by using powers and exponents and it is called a condensed writing of the prime factorization.
• Second Note: 23 = 2 × 2 × 2 = 8. 2 is called the base and 3 is the exponent. The exponent indicates how many times the base is multiplied by itself. 23 is the power and 8 is the value of the power. We sometimes say that the number 2 was raised to the power of 3.
• 6 is divisible by 6, 3, 2 and 1, so 6 is not a prime number, it is a composite number. The prime factorization of 6 = 2 × 3
• 8 is divisible by 8, 4, 2 and 1, so 8 is not a prime number, it's a composite number. The prime factorization is 8 = 23
• 9 is divisible by 9, 3, and 1, so 9 is not a prime number, it's a composite number. Its prime factorization: 9 = 32
• ### 4. Remarks on the prime numbers

• The list of the first prime numbers, up to 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
• The prime numbers are the basic building blocks of all the numbers, taking into consideration that every number can be written as a product of one or more primes. Every composite number can be written as a product of at least two prime numbers.
• EUCLID (300 B.C.) proved that as the set of natural or integer numbers is infinite, also the the set of prime numbers is infinite, with no largest prime number.
• There is no known simple formula that sets apart all of the prime numbers from the composite ones.