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Counting Numbers
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Beginning with 1, they continue infinitely in the positive direction (i.e. 1, 2, 3, 4, 5 and so on).
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Integers
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Also referred to as whole numbers, integers are the counting numbers together with their negatives (and zero). They continue infinitely in both the negative and positive direction (i.e. … -4, -3, -2, -1, 0, 1, 2, 3, 4 and so on).
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Decimals
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The decimal system allows us to write numbers that are arbitrarily small. We can represent numbers or parts of a number that are less than 1 by using a decimal. The place values to the left of the decimal represent tenths, hundredths, thousandths, and so on (i.e. 1.2, 1.003, 0.00234, and 1.235).
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Rational Numbers
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Rational numbers can be written as a ratio or fraction involving two integers. Any number that can be written as a terminating or a repeating decimal is a rational number (i.e. ½, 0.033, and 2).
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Irrational Numbers
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Irrational numbers consist of any real numbers that are not rational numbers; that is, they cannot be written as a ratio of two integers (i.e. √2, √3 , and π).
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Real Numbers
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All numbers, both rational and irrational.
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Positive Numbers
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A positive number is any number that is greater than zero (i.e. 2, √3 and ½). Non-positive numbers include zero and all negative numbers.
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Negative Numbers
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A negative number is any number that is less than zero (i.e. -2, -0.025 and -½). Non-negative numbers include zero and all positive numbers.
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Multiples
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A multiple of a number is any product of that number and an integer. For example, 4, 8, 12 and 16 are multiples of 4 because 4 × 1 = 4, 4 × 2 = 8, 4 × 3 = 12 and 4 × 4 = 16. Multiples can be positive or negative, although the test writers typically focus on positive multiples. Note: Zero is a multiple of every number since zero is an integer and any number × 0 = 0.
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Least Common Multiples
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The least common multiple is simply the lowest positive multiple shared by two numbers. For example, the multiples of 3 are 3, 6, 9, 12, 15, 18 21, 24, 27, 30, 33 etc. The multiples of 8 are 8, 16, 24, 32, 40, and so on. Therefore, the lowest positive multiple of 3 and 8 is 24.
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Factors
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The factor of a number is any positive integer that evenly divides into the number (meaning there is no remainder). For example, 18 has factors of 2, 3, 6 and 9.
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Greatest Common Factors
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The greatest common factor is the largest factor shared by two numbers. For example, the factors of 21 are 1, 3, 7 and 21. The factors of 28 are 1, 2, 4, 7, 14 and 28. Therefore, the greatest common factor of 21 and 28 is 7.
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Prime Numbers
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A prime number is any integer greater than 1 whose only factors are 1 and itself. Note, every integer has 1 and itself as a factor. Two (2) is the lowest prime number and the only even prime number (every other even number has 2 as a factor!). Other prime numbers are 3, 7, 11 and 17. Note that 1 is not a prime number. Every positive integer can be written as a unique product of prime numbers, this is called the prime factorization of a number. For example, the prime factorization of 4,620 is 2 × 2 × 3 × 5 × 7 × 11.
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Rules of Divisibility
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These rules may come in handy, especially if you need to do factoring, but remember you’ll also have your calculator:
- 1: One (1) is a factor of any whole number.
- 2: Any even number (ends in 0, 2, 4, 6 or 8) will be divisible by 2.
- 3: If the sum of all the digits in a number is divisible by 3, then the number itself is divisible by 3 (i.e. 279 is divisible by 3, but 1,246 is not).
- 4: If the last two digits of a number are divisible by 4, then the number itself is divisible by 4 (i.e. 1,236,028 is divisible by 4, but 1,844,039 is not).
- 5: Any number ending in 0 or 5 is divisible by 5.
- 6: Any even number that is divisible by 3 will be divisible by 6.
- 7*: If you double the last digit and subtract it from the rest of the number and that number is divisible by 0 or 7, than that number is divisible by 7.
- 8*: If the last three digits, taken as a number, are divisible by 8 than the whole number is divisible by 8.
- 9: This rule is similar to rule 3 above. If the sum of the digits is divisible by 9, then the number is divisible by 9.
- 10: Any number ending in 0 is divisible by 10.
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