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Test PrepACTMathNumber Properties

ACT Math Skill Review: Number Properties

For the majority of the mathematical reasoning questions, obtaining the correct answer will depend on your ability to manipulate numerical values. Approximately eighty percent (80%) of your answers will be a numerical value. Here, we will review the types of numbers that often appear on the ACT and some of their basic properties.

Classes and Types of Numbers

Yes, numbers do come in different types! Here’s a quick review of the basic terminology used.

Counting Numbers

Beginning with 1, they continue infinitely in the positive direction (i.e. 1, 2, 3, 4, 5 and so on).

Integers

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).

Decimals

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).

Rational Numbers

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).

Irrational Numbers

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).

Real Numbers

All numbers, both rational and irrational.

Positive Numbers

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.

Negative Numbers

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.

Multiples

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.

Least Common Multiples

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.

Factors

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.

Greatest Common Factors

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.

Prime Numbers

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.

Rules of Divisibility

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.

*These rules are not particularly helpful and, thus, are not worth remembering.

Zero

Zero has a few unique properties that make it worth its own section. Here’s a quick rundown of what makes zero so special:
  • It is neither negative nor positive.
  • It is even.
  • Any number multiplied by 0 equals 0.
  • Zero divided by any number equals 0. (Note: You cannot divide by 0.)
  • Any number plus 0 equals the number. Translation: Adding zero doesn’t change anything.
  • Any number raised to the 0 power equals 1. Thus 2^0=1. Zero raised to any power equals 0.

Examples


 

Answers and Explanations

  1. The correct answer is B. The only thing we know for sure is that none of the values for a, b, c, d and e can be 0. It’s important to recognize that the question is asking which MUST be true, not which COULD be true. Choices A, C, D could be true, while choice E must be false. Choice B is the only one that must be true.
     
  2. The correct answer is H. First off, to even consider the truth of the statement, you need a number that is divisible by both 3 and 7. Only choices H and J fit this requirement. Therefore, you can eliminate choices F, G and K. Choice J is divisible by 3 (goes into 105 35 times), 7 (goes into 105 15 times) AND 5 (goes into 105 21 times), so rather than disproving the statement, it would appear to support it. Choice H however, provides a number (21) that, while divisible by 3 and 7, is not divisible by 5. Thus, it disproves the statement.


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