1) Number Properties

i) Integers

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.

iii) Factors

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.

v) Multiples

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.

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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

Function

Result

even + even

even

even + odd

odd

odd + odd

even

even - even

even

even - odd

odd

odd - odd

even

even * even

even

even * odd

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

Note:

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.

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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

Function

Result

positive * positive

positive

positive * negative

negative

negative * negative

positive

positive ÷ positive

positive

positive ÷ negative

negative

negative ÷ negative

positive

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...