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.
So the factors of 6 are: 1, 2, 3, and 6.
Any factors that are the same for two (or more) numbers are called Common Factors.
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 SAT 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.
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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The beauty of inequalities on the SAT is that they allow you to do all of the same things that you can with equations:
For some reason, the SAT is absolutely obsessed with percentage problems, and as a result, many students have trouble. The following set of formulas will help eradicate your trouble once and for all.
Math fundamentals deal with number properties, their classifications and basic operations on numbers.
Some strategies are so powerful that they can singlehandedly increase your SAT Math Test score by more than 100 points. At AP Guru, we absolutely recommend students use these three strategies time and again.
You’re allowed to use a calculator on Section 4 of the SAT, but remember to use it as a tool, not as a crutch.
The SAT Math Test contains one 55-minute section with 38 questions (calculators ok) and one 25-minute section with 20 questions (calculators not permitted).