# Prime Numbers. Mathematical Operations With Prime Factors

## Prime or composite numbers? The last 3 numbers that have been decomposed into prime factors (prime factorized)

 Is 2,056,628,753 a prime number or a composite one? What is the prime factorization of this number? Aug 11 11:51 UTC (GMT) Is 52,629,075 a prime number or a composite one? What is the prime factorization of this number? Aug 11 11:41 UTC (GMT) Is 951 a prime number or a composite one? What is the prime factorization of this number? Aug 11 11:41 UTC (GMT) » New. All the Calculations Performed by Our Visitors: Prime Numbers or Not? Prime Factorizations of Composite Numbers. Data organized on a Monthly Basis

## 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 374 and 153? How to calculate the GCF (HCF, GCD)? Aug 11 11:51 UTC (GMT) What is the greatest (highest) common factor (divisor) of the numbers 4,915 and 7,140? How to calculate the GCF (HCF, GCD)? Aug 11 11:49 UTC (GMT) What is the greatest (highest) common factor (divisor) of the numbers 209 and 264? How to calculate the GCF (HCF, GCD)? Aug 11 11:49 UTC (GMT) » New. All the Calculations Performed by Our Visitors: Calculated Values of the Greatest (Highest) Common Factor (Divisor), gcf, hcf, gcd, of Pairs of Numbers. Data organized on a Monthly Basis

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

 What is the least (the lowest) common multiple, LCM, of the numbers 19 and 11 and how to calculate it? Aug 11 11:50 UTC (GMT) What is the least (the lowest) common multiple, LCM, of the numbers 474 and 2,842 and how to calculate it? Aug 11 11:49 UTC (GMT) What is the least (the lowest) common multiple, LCM, of the numbers 45 and 20 and how to calculate it? Aug 11 11:46 UTC (GMT) » New. All the Calculations Performed by Our Visitors: Calculated Values of the Least (the Lowest) Common Multiple, lcm, of Pairs of Numbers. Data organized on a Monthly Basis

## 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 36 / 15 to the lowest terms (to its simplest form, irreducible - the smallest possible numerator and denominator) Aug 11 11:44 UTC (GMT) Completely reduce (simplify) the fraction 7 / 25 to the lowest terms (to its simplest form, irreducible - the smallest possible numerator and denominator) Aug 11 10:59 UTC (GMT) Completely reduce (simplify) the fraction 7 / 25 to the lowest terms (to its simplest form, irreducible - the smallest possible numerator and denominator) Aug 11 10:58 UTC (GMT) » New. All the Calculations Performed by Our Visitors: Fractions reduced (simplified) to their lowest terms (simplest form). Data organized on a Monthly Basis

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

 Is the number 486 divisible by 3? Could 486 be evenly divided by 3? Does the first number contain all the prime factors of the second? Aug 11 11:50 UTC (GMT) Is the number 486 divisible by 3? Could 486 be evenly divided by 3? Does the first number contain all the prime factors of the second? Aug 11 11:49 UTC (GMT) Is the number 486 divisible by 3? Could 486 be evenly divided by 3? Does the first number contain all the prime factors of the second? Aug 11 11:49 UTC (GMT) » New: All the Calculations Performed by Our Visitors: Numbers that were checked for divisibility. Data organized on a Monthly Basis

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

 What are all the proper, improper and prime factors (all the divisors) of the number 81,668,162? How to calculate them? Aug 11 11:48 UTC (GMT) What are all the proper, improper and prime factors (all the divisors) of the number 1,623,787? How to calculate them? Aug 11 11:47 UTC (GMT) What are all the proper, improper and prime factors (all the divisors) of the number 264,172? How to calculate them? Aug 11 11:45 UTC (GMT) » New: All the Calculations Performed by Our Visitors: Common Divisors of Numbers. Data organized on a Monthly Basis

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

 Are the two numbers 19 and 57 coprime (relatively prime, prime to each other) or not? Aug 11 11:50 UTC (GMT) Are the two numbers 1,203 and 250 coprime (relatively prime, prime to each other) or not? Aug 11 11:39 UTC (GMT) Are the two numbers 196,382 and 5,956 coprime (relatively prime, prime to each other) or not? Aug 11 11:35 UTC (GMT) » New: All the Calculations Performed by Our Visitors: Coprime Numbers (Prime to Each Other Numbers)? Data organized on a Monthly Basis

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

 Numbers parity: Is 3,123,282 an even or an odd number? Aug 11 11:50 UTC (GMT) Numbers parity: Is 3,123,282 an even or an odd number? Aug 11 11:50 UTC (GMT) Numbers parity: Is 3,123,282 an even or an odd number? Aug 11 11:50 UTC (GMT) » New: All the Calculations Performed by Our Visitors: Even or Odd Numbers? Data organized on a Monthly Basis

## 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.