Natural Numbers and Integers

Order of operations, brackets as grouping symbols
The divisibility of integers
Prime and composite numbers
The greatest common divisor/factor (GCD/GCF)
The least common multiple (LCM)
Order of operations - brackets as grouping symbols
When expressions have more than one operation, we have to follow rules for the order of operations. First do all multiplication and division operations working from left to right. Next do all addition and subtraction.
 Example: - 8 -  3 · (- 4) + 15 ¸ (- 5) = - 8 - (- 12) + (- 3) = - 8 + 12 - 3 = 1
We often use grouping symbols, like brackets, to help us organize complicated expressions into simpler ones. Do operations in brackets and other grouping symbols first. If there are grouping symbols within other grouping symbols do the innermost first.
 Example: - 3 + 2 · [- 2 - 3 · (- 4 ¸ 2 + 1)] = - 3 + 2 · [- 2 - 3 · (- 2 + 1)] = - 3 + 2 · [- 2 - 3 · (- 1)] = = - 3 + 2 · [- 2 + 3] = - 3 + 2 = - 1
The divisibility of integers
A divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. Numbers divisible by 2 are called even and those that are not are called odd.
 Example: 7 is divisor of 21 because 21 ¸ 7 = 3. We also say 21 is divisible by 7 or 21 is a multiple of 7. The divisors of  21 are  { 1, 3, 7, 21 }.
The divisibility rules
 A number is divisible by 2 if the last digit is divisible by 2. A number is divisible by 3 if the sum of its digits is divisible by 3. A number is divisible by 4 if the number given by the last two digits is divisible by 4. A number is divisible by 5 if the last digit is 0 or 5. A number is divisible by 6 if it is divisible by 2 and by 3. A number is divisible by 7 if the result of subtracting twice the last digit from the number (with the last  digit removed) is divisible by 7 (e.g. 581 is divisible by 7 since 58 - 2 · 1 = 56 is divisible by 7). A number is divisible by 8 if the number given by the last three digits is divisible by 8. A number is divisible by 9 if the sum of its digits is divisible by 9. A number is divisible by 10 if the last digit is 0.
Prime and composite numbers
A prime number is a natural number greater than one whose only positive divisors are one and itself.
A natural number that is greater than one and is not a prime is called a composite number. The sequence of prime numbers begins 2, 3, 5, 7, 11, 13, 17, 19, 23, ...
The integer prime-factorization
 Every positive integer greater than 1 can be factored as a product of prime numbers. Given a number n divide by the first prime from the list of primes, if it does not divide cleanly, divide n by the next prime, and so on. Example: Thus, the prime factors of  350 = 2 · 5 · 5 · 7.
 350 | 2 175 | 5 35 | 5 7 | 7 1 |
The greatest common divisor (GCD)
The greatest common divisor of two integers is the largest integer that divides both numbers.
Greatest common divisors can be computed by determining the prime factors of the two numbers and comparing the factors, as in the following example:
 Example: 72 = 2 · 2 · 2 · 3 · 3,   90 = 2 · 3 · 3 · 5,    =>     GCD(72, 90) = 2 · 3 · 3 = 18.
Two numbers are called relatively prime if their greatest common divisor equals 1. Thus, for example
12 = 2 · 2 · 3,   25 = 5 · 5    =>    GCD(12, 25) = 1.
The least common multiple (LCM)
A common multiple is a number that is a multiple of two or more numbers.
The least common multiple of two integers a and b is the smallest positive integer that is a multiple of both a and b. To find the least common multiple of two numbers: - first list the prime factors of each number, then multiply each factor the greatest number of times it occurs in either number. If the same factor occurs more than once in both numbers, multiply the factor the greatest number of times it occurs.
 Example: Let's find the LCM of 24, 54 and 60,
the prime factors:  24 = 2 · 2 · 2 · 3,    54 = 2 · 3 · 3 · 3,    60 = 2 · 2 · 3 · 5,
==>    LCM(24, 54, 60) = 2 · 2 · 2 · 3 · 3 · 3 · 5 = 1080.
Beginning Algebra Contents A