The divisibility of the natural numbers:

Method 1: Divide the numbers and check the remainder of the operation. If the remainder is zero, then the numbers are divisible.

Method 2: The prime factorization of the numbers (the decomposition of the numbers into prime factors).

• 1. Divisibility:

• One natural number is said to be divisible by another natural number if after dividing the two numbers, the remainder of the operation is zero.
• Example: Let's divide two different numbers: 12 and 15, by 4.
• When dividing 12 by 4, the quotient is 3 and the remainder of the operation is zero.
• But when we divide 15 by 4, the quotient is 3 and the operation leaves a remainder of 3.
• We say that the number 12 is divisible by 4 and 15 is not divisible by 4.
• We also say that 4 is a divisor, or a factor, of 12, but not a factor (divisor) of 15.

• We say that the number "a" is divisible by "b", if there is an integer number "n", such that:
• a = n × b.
• The number "b" is called a divisor (or a factor) of "a" ("n" is also a divisor, or a factor, of "a").

• 2. Some divisibility rules:

• 0 is divisible by any number other than itself.
• 1 is a divisor (a factor) of every number.
• Improper factors: Any number "a", different of zero, is divisible at least by 1 and itself. In this case the number itself, "a", is called an improper factor (or an improper divisor). Some also consider 1 as an improper factor (divisor).
• Prime numbers: A number which is divisible only by 1 and itself is also called a prime number.
• Coprime numbers: If the greatest common factor of two numbers, "m" and "n", the GCF (m; n) = 1 - then it means that the two numbers are coprime, in other words they have no divisor other than 1. If a number "a" is divisible by these two coprime numbers, "m" and "n", then "a" is also divisible by their product, (m × n).
  • Example:
  • The number 84 is divisible by 4 and 3 and is also divisible by 4 × 3 = 12.
  • This is true because the two divisors, 3 and 4, are coprime.

• 3. Calculating the divisors (factors):

• Calculating the divisors (factors) of a number is very useful when simplifying fractions (reducing fractions to lower terms).
• The established rules for finding factors (divisors) are based on the fact that the numbers are written in the decimal system:
• Multiples of 10 are divisible by 2 and 5, because 10 is divisible by 2 and 5
• Multiples of 100 are divisible by 4 and 25, because 100 is divisible by 4 and 25
• Multiples of 1,000 are divisible by 8, because 1,000 is divisible by 8.
• All the powers of 10, when divided by 3, or 9, have a remainder equal to 1.

• Due to the rules of operations with remainders, we have the following remainders when dividing numbers by 3 or 9:
• 600 leaves a remainder equal to 6 = 1 × 6 (1 for every 100)
• 240 = 2 × 100 + 4 × 10, then the remainder will be equal to 2 × 1 + 4 × 1 = 6

• When a number is divided by 3 or 9, the remainder is equal to what you get by dividing the sum of the digits of that number by 3 or 9:
• 7,309 has the sum of its digits: 7 + 3 + 0 + 9 = 19, which is divided with a remainder by either 3 or 9. So 7,309 is divisible by neither 3 nor by 9.

• All the even powers of 10, such as 10² = 100, 10⁴ = 10,000, 10⁶ = 1,000,000, and so on, when divided by 11 have a remainder equal to 1.
• All the odd powers of 10, such as 10¹ = 10, 10³ = 1,000, 10⁵ = 100,000, 10⁷ = 10,000,000, and so on, when divided by 11 have a remainder equal to 10. In this case, the alternating sum of the digits of the number has the same remainder as the number itself when divided by 11.
• How is the alternating sum of the digits being calculated - it is shown in the example below.

• For instance, for the number: 85,976: 6 + 9 + 8 = 23, 7 + 5 = 12, the alternating sum of the digits: 23 - 12 = 11. So 85,976 is divisible by 11.

• 4. Quick ways to determine whether a number is divisible by another one or not:

• 2, if the last digit is divisible by 2. If the last digit of a number is 0, 2, 4, 6 or 8, then the number is divisible by 2. For example, the number 20: 0 is divisible by 2, so then 20 must be divisible by 2 (indeed: 20 = 2 × 10).
• 3, if the sum of the digits of the number is divisible by 3. For example, the number 126: the sum of the digits is 1 + 2 + 6 = 9, which is divisible by 3. Then the number 126 must also be divisible by 3 (indeed: 126 = 3 × 42).
• 4, if the last two digits of the number make up a number that is divisible by 4. For example 124: 24 is divisible by 4 (24 = 4 × 6), so 124 is also divisible by 4 (indeed: 124 = 4 × 31).
• 5, if the last digit is divisible by 5 (the last digit is 0 or 5). For example 100: the last digit, 0, is divisible by 5, then the number 100 must be divisible by 5 (indeed: 100 = 5 × 20).
• 6, if the number is divisible by both 2 and 3. For example, the number 24 is divisible by 2 (24 = 2 × 12) and is also divisible by 3 (24 = 3 × 8), then it must be divisible by 6. Indeed, 24 = 6 × 4.
• 7, if the last digit of the number (the unit digit), doubled, subtracted from the number made up of the rest of the digits gives a number that is divisible by 7. The process can be repeated until a smaller number is obtained. For example, is the number 294 divisible by 7? We apply the algorithm: 29 - (2 × 4) = 29 - 8 = 21. 21 is divisible by 7. 21 = 7 × 3. But we could have applied the algorithm again, this time on the number 21: 2 - (2 × 1) = 2 - 2 = 0. Zero is divisible by 7, so 21 must be divisible by 7. If 21 is divisible by 7, then 294 must be divisible by 7.
• 8, if the last three digits of the number are making up a number that is divisible by 8. For example, the number 2,120: 120 is divisible by 8 since 120 = 8 × 15. Then 2,120 must also be divisible by 8. Proof: if we divide the numbers, 2,120 = 8 × 265.
• 9, if the sum of the digits of the number is divisible by 9. For example, the number 270 has the sum of the digits equal to 2 + 7 + 0 = 9, which is divisible by 9. Then 270 must also be divisible by 9. Indeed: 270 = 9 × 30.
• 10, if the last digit of the number is 0. Example, 140 is divisible by 10, since 140 = 10 × 14.
• 11 if the alternating sum of the digits is divisible by 11. For example, the number 2,915 has the alternating sum of the digits equal to: (5 + 9) - (1 + 2) = 14 - 3 = 11, which is divisible by 11. Then the number 2,915 must also be divisible by 11: 2,915 = 11 × 265.
• 25, if the last two digits of the number are making up a number that is divisible by 25. For example, the number made up by the last two digits of the number 275 is 75, which is divisible by 25, since 75 = 25 × 3. Then 275 must also be divisible by 25: 275 = 25 × 11.

What is a prime number? Definition, examples

What is a composite number? Definition, examples

The prime numbers up to 1,000

The prime numbers up to 10,000

The Sieve of Eratosthenes

The Euclidean Algorithm

Completely reduce (simplify) fractions to the lowest terms: Steps and Examples