Math: Integer Properties & Arithmetic
Even & Odd integers
Even integers can be divided evenly by two; odd integers cannot. Examples: 2, 4, 6, 0, -2, -4, 6 are all even. 1, 3, 5, -1, -3, -5 are all odd.
Positive & Negative numbers
Positive numbers are greater than 0 and negative numbers are less than 0. 0 is neither positive nor negative! Example: -1, -13, and -14,560 are negative numbers. 1, 13, and 14,560 are positive numbers
Multiples
The result of multiplying a certain number by an integer. Multiples can be positive OR negative.When you learned your times tables, you were learning multiples. 4, 6, 8, and 10 are multiples of 2 because 2 × 2 = 4, 2 × 3 = 6, 2 × 4 = 8 and so on.
Perfect squares
A perfect square is a number that can be expressed as the product of two equal integers. Examples: 1: (1 × 1), 4: (2 × 2), 9: (3 × 3), 81: (9 x 9), 625: (25 × 25)
Remainders
A remainder is the amount "left over" after dividing one integer by another to produce an integer quotient.
Absolute value
Absolute value is the distance on the number line from a given number to zero. To find the absolute value of a number, simply remove the negative sign in front of a number. Absolute value notation looks like this: |-2|. Example: the absolute value of -15, or |-15|, is 15.
Operations of Even & Odd integers
Adding, subtracting, multiplying, and dividing even and odd numbers always yields the same results regarding whether the result is odd or even. For example, even + even = even and even + odd = odd. Important to Know! If this comes up on the ACT, simply test an easy case. Would an odd - odd be even or odd? Try 5 - 3 , which equals 2, so odd - odd is always even.
Prime numbers
An integer greater than 1 that has no positive divisors other than 1 and itself. Important to know! 1 is NOT a prime number. 2 IS a prime number. The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.
Order of Operations (PEMDAS)
PEMDAS is an acronym that helps you remember the order you must do operations in on an arithmetic problem: 1. Parentheses (any grouping symbols, including brackets and fraction bars) 2. Exponents 3. Multiplication 4. Division 5. Addition 6. Subtraction. Good to know! You may remember this better as "Please Excuse My Dear Aunt Sally."
Rounding
Rounding is a method used to make numbers shorter and simpler by leaving off some of digits of smaller values. For the digit you are rounding, if it is less than 5, round down and equal to or greater than 5, round up. Examples: 23,459 rounded to the thousands place is 23,000 and rounded to the hundreds place is 23,500. 0.089 rounded to the tenths place is 0.1.
Scientific notation
Scientific notation is a method to handle very large or very small numbers without writing out all the place holding zeros. Simply count how many spaces you are moving the decimal point to the right or the left. There should always be one digit before the decimal point. Examples: 5,340,000,000 is 5.34 × 10^9 and 0.0000000000425 is 4.25 ×10^-11
Greatest common factor
The greatest factor that divides two numbers. To find the GCF, list the prime factors of each number, circle the pairs of factors both numbers have in common, and multiply those factors together. Example: The GCF of 12 (2 × 2 × 3) and 30 (2 × 3 × 5) is 6 (2 x 3). To check, make sure both 12 and 30 divide by 6.
Prime factors
The prime factors of a number are the prime numbers that are divisors of that number. Examples: The prime factors of 15 are 3 and 5. The prime factors of 18 are 3 and 2 because 3 ×3 × 2 = 18.
Least common multiple
The smallest positive integer that is divisible by two numbers. To find the LCM: Method 1: list the prime factors of each number and multiply each factor by the greatest number of times it occurs in either number. Method 2: list the multiples of a number and find the first one that is the same. Example for Method 1: The LCM of 8 (2 × 2 × 2) and 14 (2 × 7) is 56 (2 × 2 × 2 × 7). Example for Method 2: 8: 8, 16, 24, 32, 40, 48, 56. 14: 14, 28, 42, 56. So the LCM is 56.
