Number Properties

Math fundamentals deal with number properties, their classifications and basic operations on numbers. Here are the types of numbers used most often:

  1. REAL NUMBERS: Any number that is possible including negatives and fractions. Cannot be irrational EXAMPLE: 0, 1, -2, 1/2, \(\sqrt{25}\).....
  2. WHOLE NUMBERS: Any number without fractions or decimals. Cannot be negative. EXAMPLE: 0, 1, 2, 3, 4...
  3. NATURAL NUMBER: Any number that is a positive whole number EXAMPLE: 1, 2, 3, 4...
  4. INTEGERS: Any whole numbers and negatives of any whole numbers. EXAMPLE: ... -4, -3, -2, -1, 0, 1, 2, 3, 4 ...
  5. RATIONAL NUMBER: Any number that can be written by dividing one integer by another - in plain English, any number that can be written as a fraction or ratio. (An easy way to remember this is to think of rational root word “ratio.”) EXAMPLE: 1/2, 2.5, 4.12, 1/3
  6. IRRATIONAL NUMBER: Any number that cannot be written as a simple fraction (because the decimal goes on forever without repeating). EXAMPLE: 3.14159265... , \(\sqrt{-2}\)

Every number has a decimal expansion. For example, 2 can be written 2.000... However, you can easily spot an irrational number because the decimal expansion goes on forever without repeating. Here’s how all types of numbers fit together:

Example:

  • -2 is an integer, rational and a real number
  • 6.675 is rational and real. (terminating decimals or decimals that end are rational.)
  • \(\sqrt{5}\) = 2.2360679775 is irrational and real. (Non repeating decimals that go on forever are irrational.)
Solved Example 1 
If a is an odd integer and b is an even integer, which of the following is an odd integer?
A.
3b
B. a + 3
C. 2(a + b)
D. a + 2b

Solution:
Answer choice D is correct. Twice b will be even because an even number x an even number = an even number. And the final result will be odd because of an odd number (a) + an even number (2b) = an odd number.

You could also solve this by double-checking these rules by using your own numbers.
If you assign an odd number to a and an even number to b, you can test out each option in about the same amount of time.

So, for this question, you could have said a was 3, and b was 4. Then option D would have looked like this: 3 + 2(4) = 11. D is the correct answer.

Factors

Factors are integers you multiply together to get another integer. When finding the factors of a number, ask yourself, “What numbers can be multiplied together to give me this number?” Every number greater than 1 has at least two factors because every number can be divided by 1 and itself.

Example: What are the factors of 6?

  • 2 and 3 are factors of 6, because 2 x 3 = 6.
  • 1 and 6 are also factors of 6, because 1 x 6 = 6.

So the factors of 6 are: 1, 2, 3, and 6.

Shortcuts to find factors of some integers:

  • An integer is divisible by 2 if it ends in an even number. EXAMPLE: 10, 92, 44, 26, and 8 are all divisible by 2 because they end in an even number.
  • An integer is divisible by 3 if the same of its digits is divisible by 3. EXAMPLE: 42 is divisible by 3 because 4 + 2 = 6, and 6 is divisible by 3.
  • An integer is divisible by 5 if it ends in 0 or 5. EXAMPLE: 10, 65, and 2,320 are all divisible by 5 because they end in either 0 or 5.
  • An integer is divisible by 9 is the same of the digits is divisible by 9. EXAMPLE: 297 is divisible by 9 because 2 + 9 + 7 = 18, and 18 is divisible by 9.

Any factors that are the same for two (or more) numbers are called Common Factors.

Example: What are the common factors of 12 and 18?

  • The factors for 12 are 1, 2, 3, 4, 6, 12.
  • The factors for 18 are 1, 2, 3, 6, 9, 18.

The common factors of 12 and 18 are 1, 2, 3, and 6. The largest factor that both numbers share is called the GREATEST COMMON FACTOR, or GCF for short. The GCF of 12 and 18 is 6.

Multiples and LCM

When we multiply a number by any whole number (that isn’t 0), the product is a multiple of that number. Every number has an infinite list of multiples.

Any multiples that are the same for two (or more) numbers are called Common Multiples.

What is the smallest multiple that both 2 and 5 have in common? The smallest multiple is 10. We call this the least common multiples, or LCM.

Example: Susie signs up to volunteer at the animal shelter every 6 days. Luisa signs up to volunteer at the shelter every 5 days. If they both sign up to volunteer on the same day, when is the first day that Susie and Luisa will work together?

This is the same as saying, "Find the LCM for 5 and 6."

Susie will work on the following days: 6th, 12th, 18th, 24th and 30th. 30 is the first number divisible by 5, so the LCM is 30. The first day that Susie and Luisa will work together is on the 30th day

Prime Numbers

The ACT loves to test students on prime numbers, so you should expect to see one question per test on prime numbers. Be sure to understand what they are and how to find them. A prime number is a number that is only divisible by two numbers - itself and 1. The only even prime number is 2.

Solved Example 2 
Questions about primes come up fairly often on the ACT and understanding that 2 (and only 2) is an even prime number will be invaluable for solving many problems.

A prime number x is squared and then added to a different prime number, y. Which of the following could be the final result?
I. An even number
II. An odd number
III. A positive number

A.
I only
B. II only
C. III only
D. I, II, and III

Solution:
This question relies on your knowledge of both number relationships and primes. You know that any number squared (the number times itself) will be an even number if the original number was even, and an odd number if the original number was odd.

Next, we are adding that square to another prime number. Knowing that 2 is a prime number, let’s replace x with 2.

2 x 2 = 4.

Now if y is a different prime number (as stipulated in the question), it must be odd, because the only even prime number is 2.

So let’s say y = 3.

4 + 3 = 7. The end result is odd.
THIS MEANS II IS CORRECT.
 

But what if both x and y were odd prime numbers? Let’s try this next and say that x = 3 and y = 5.

So, 3 x 3 = 9 and 9 + 5 = 14.

The end result is even. THIS MEANS I IS CORRECT.

Our results show that it is possible to get a positive number result as well since both our results were positive. THIS MEANS III IS CORRECT TOO. The correct answer is D. 

If you forgot that 2 was a prime number, you would have picked D, I and III only, because there would have been no possible way to get an odd number. Remembering that 2 is a prime number is key to solving this question.
Solved Example 3
How many prime numbers are between 30 and 50, inclusive?
A.
Two
B. Three
C. Four
D. Five

Solution:
This might seem intimidating or time-consuming, but you do NOT need to memorize a list of prime numbers. First, consider all numbers between 30 and 50. Eliminate all even numbers from the list, as you know the only even prime number is 2.

Next, eliminate all numbers that end in 5. Remember that prime numbers are only divisible by themselves and 1. Any number that ends in 5 or 0 is divisible by 5. So, now your list looks like this: 31, 33, 37, 39, 41, 43, 47, 49.

This smaller list is much easier to work with, but we need to narrow it down even further. First, start eliminating numbers divisible by 3: 33 and 39 are gone.

We are left with 31, 37, 41, 43, 47, 49.

Eyeball each number to see if it has any factors. You can eliminate 49 since it has a factor 7.

You are left with 5 prime numbers: 31, 37, 41, 43, and 47. The correct answer is D.

Exponents

An exponent is the number of times the base number is multiplied by itself. Things to remember about exponents:

1. Any base without an exponent has an "invisible" exponent of 1.
Example: 8 = \(8^{1}\)

2. Any base with an exponent 0, equals 1.
Example: \(6^{0}\) = 1

3. Be careful when calculating negative numbers with exponents.
Example: \(-3^{2}\) = -( \(3^{2}\) ) = -(3 x 3) = -9 vs ( \(-3^{2}\) )= (-3) x (-3) = 9

You can simplify expressions with more than one exponent by combining the exponents - the only requirement is that the base must be the same. It looks like this:

\(x^{a}\) x \(x^{b}\) = \(x^{a+b}\)

\(x^{a}\)  \(x^{b}\) = \(x^{a+b}\)

4. When there is an exponent inside the parenthesis and another outside the parentheses, this is called a POWER OF A POWER. A power of a power can be simplified by multiplying the exponents. It looks like this:

\((v^{a})^{b}\) = \(v^{axb}\)

Example: \((4^{2})^{3}\)  = \(4^{2x3}\) = \(4^{6}\)

A negative exponent in the numerator becomes a positive exponent when moved to the denominator. It looks like this:

\[x^{m}=\frac{1}{x^{m}}\]

Example: Solve for X in \(4^{5}\) = \(8^{5-x}\)

Since the bases are different, we need to work some magic to rewrite the equation so the bases are the same.

4 = \(2^{2}\) and 8 = \(2^{3}\)

Substitute \(2^{2}\) and \(2^{3}\) into the original equation \((2^{2})^{5}=(2^{3})^{6-x}\)

Using the “power to a power” rule, we simplify: \(2^{10}=2^{18-3x}\)

Like before, drop the bases and make an equation with the exponents.

10 = 18 - 3x
3x = 8
x = 8/3

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