1) Number Properties
Numbers, such as -1, 0, 1, 2, and 3, that have no fractional part. Integers include the counting numbers (1, 2, 3, …), their negative counterparts (-1, -2, -3, …), and 0.
ii) Whole & Natural Numbers
The terms from 0,1,2,3,….. are known as Whole numbers. Natural numbers do not include 0.
Positive integers that divide evenly into an integer. Factors are equal to or smaller than the integer in question. 12 is a factor of 12, as are 1, 2, 3, 4, and 6. iv) Factor Foundation Rule
If a is a factor of b, and b is a factor of c, then a is also a factor of c. For example, 3 is a factor of 9 and 9 is a factor of 81. Therefore, 3 is also a factor of 81.
Multiples are integers formed by multiplying some integer by any other integer. For example, 6 is a multiple of 3 (2 * 3), as are 12 (4 * 3), 18 (6 * 3), etc. In addition 3 is also a multiple of itself i.e. 3 (1*3). Think of multiples as equal to or larger than the integer in question
vi) Prime Numbers
A positive integer with exactly two factors: 1 and itself. The number 1 does not qualify as prime because it has only one factor, not two. The number 2 is the smallest prime number; it is also the only even prime number. The numbers 2, 3, 5, 7, 11, 13 etc. are prime.
vii) Prime Factorization
Prime factorization is a way to express any number as a product of prime numbers. For example, the prime factorization of 30 is 2 * 3 * 5. Prime factorization is useful in answering questions about divisibility.
viii) Greatest Common Factor
Greatest Common FACTOR refers to the largest factor of two (or more) integers. Factors will be equal to or smaller than the starting integers. The GCF of 12 and 30 is 6 because 6 is the largest number that goes into both 12 and 30.
viii) Least Common Multiple (LCM)
Least Common Multiple refers to the smallest multiple of two (or more) integers. Multiples will be equal to or larger than the starting integers. The LCM of 6 and 15 is 30 because 30 is the smallest number that both 6 and 15 go into.
ix) Odd & Even Numbers
Any number divisible by 2 is even and not divisible by 2 is odd. Odd & Even number Rules
even + even
even + odd
odd + odd
even - even
even - odd
odd - odd
even * even
even * odd
odd * odd
even ÷ even
anything (even, odd, or not an integer)
even ÷ odd
even or not an integer
odd ÷ even
not an integer
odd ÷ odd
odd or not an integer
Division rules are more complicated because an integer answer is not always guaranteed. If the result of the division is not an integer, then that result cannot be classified as either even or odd.
x) Absolute Value
The distance from zero on the number line. A positive number is already in the same form as that number‟s absolute value. Remove the negative sign from a negative number in order to get that number‟s absolute value. For example the absolute value of - 2 is 2.
xi) Positive-Negative Number Rules
positive * positive
positive * negative
negative * negative
positive ÷ positive
positive ÷ negative
negative ÷ negative
xii) Product of n consecutive integers and divisibility
The product of n consecutive integers is always divisible by n! Given 5*6*7*8, we have n = 4 consecutive integers. The product of 5*6*7*8 (=1680), therefore, is divisible by 4! = 4*3*2*1 = 24.
xiii) Sum of n consecutive integers and divisibility
There are two cases, depending upon whether n is odd or even:
If n is odd, the sum of the integers is always divisible by n. Given 5+6+7, we have n = 3 consecutive integers. The sum of 5+6+7 (=18), therefore, is divisible by 3.
If n is even, the sum of the integers is never...
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