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Number Properties

**REAL NUMBERS:**Any number that is possible including negatives and fractions. Cannot be irrational EXAMPLE: 0, 1, -2, 1/2, \(\sqrt{25}\).....**WHOLE NUMBERS:**Any number without fractions or decimals. Cannot be negative. EXAMPLE: 0, 1, 2, 3, 4...**NATURAL NUMBER:**Any number that is a positive whole number EXAMPLE: 1, 2, 3, 4...**INTEGERS:**Any whole numbers and negatives of any whole numbers. EXAMPLE: ... -4, -3, -2, -1, 0, 1, 2, 3, 4 ...**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**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:**

- -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.)

If a is an odd integer and b is an even integer, which of the following is an odd integer?

A.

Solution:

You could also solve this by double-checking these rules by using your own numbers.

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.

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.

- 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.

- 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**.* ***

- 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.

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

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.

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.

Solution:

Next, we are adding that square to another prime number. Knowing that 2 is a prime number,

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.

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.

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.

How many prime numbers are between 30 and 50, inclusive?

A.

Solution:

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.

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.

** 2. **Any base with an exponent 0, equals 1.

** 3. **Be careful when calculating negative numbers with exponents.

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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