Refresher - Basic Properties of Natural Numbers

Basic Properties of Natural numbers

Number Theory is the branch of mathematics that explores the properties of integers, primarily natural numbers. Number theory is at the heart of secure messaging systems and cryptography.

The counting numbers 1, 2, 3, . . . together with their negatives and zero make up the set of integers. Multiples of 2 are integers called the even numbers. These include –2, 0, 2, 4,….. The odd numbers are those not divisible by 2 such as –1, 1, 3 ……

The addition and multiplication of integers will produce another integer. Dividing an integer by another, however, does not always produce an integer value.

Divisibility

 If the number a is divisible by b, we can use the notation: b | a (“b divides a”) and then the operation a ÷ b produces an integer with no remainder. We say that b is a divisor or factor of a, and a is a multiple of b.

The number 1 divides all integers. Example: $\frac{5}{1}=5$

Any nonzero number divided by itself is equal to 1. Example: $\frac{2}{2}=1$

Basic Divisibility Rules

A divisibility rule is a method that can be used to determine whether a number is evenly divisible by other numbers. Divisibility rules are a shortcut for testing a number's factors without resorting to division calculations. (Wikipedia)

Dividing by 2: All even numbers are divisible by 2.
For example: all numbers ending in 0, 2, 4, 6 or 8.

Dividing by 3: If the sum of the digits is divisible by 3, the original number was divisible by 3.
For example: 531 (5+3+1=9) 9 is divisible by 3, 531 is divisible by 3

Dividing by 4: If the last two digits in a number is divisible by 4, the number is divisible by 4.
For example: 328 ends in 28 which is divisible by 4, 328 is divisible by 4.

Dividing by 5: If a number ends in a 5 or a 0, it is divisible by 5.
For example: 500, 2000, 125, 9035

Dividing by 6: If a number is divisible by 2 and 3, it is divisible by 6.
For example: 150 ÷ 2 = 75 and 150 ÷ 3 = 50, 150 is divisible by 6

Dividing by 7: If the last digit in a number is doubled and subtracted from the rest of the digits and the difference is divisible by 7, the number is divisible by 7.
For example: 252: double the 2 to get 4, subtract 4 from 25 = 21, 21 is divisible by 7, 252 is divisible by 7

Dividing by 8: If the last 3 digits are divisible by 8, the number is divisible by 8.
Example: 2120 - The last 3 digits, 120 are divisible by 8, 2120 is divisible by 8

Dividing by 9: If the sum of the digits is divisible by 9, the number is divisible by 9.
For example: 331830, (3 + 3 + 1 + 8 + 3 + 0= 18), 18 is divisible by 9

Dividing by 10: If the number ends in a 0, the number is divisible by 10.
For example: 560

Number theory includes the study of perfect numbers. Perfect numbers are those positive integers which are equal to the sum of all their factors (not including the number itself). The smallest example is 6 (6 = 1 + 2 + 3). The first four perfect numbers, 6, 28, 496, 8,128 were known over 2,000 years ago. Pythagoras gave these four numbers their “perfect” name. Euclid proved that if ${\displaystyle n=(2^p-1)2^{p-1}}$ with ${\displaystyle (2^p-1)}$ prime then n is a perfect number.