2.4 SAT Math - Primes, Factors & more

Prime Numbers

A Prime Number is technically "A number whose only factors are itself and 1 where those two numbers are distinct." What does this mean in regular words? It means nothing goes into prime numbers.  Prime numbers have only two factors, itself and the number 1.  It also means that the number 1 is NOT prime. 

The first five prime numbers are: 2, 3, 5, 7, 11.  Know these.  Most (though certainly not all) prime number questions will deal with these first five on the SAT math section.

Two is a particularly special prime number.  Why? It's the smallest, which is good to know, but it's also the ONLY even prime number.  This means if the question requires you to try a few options in order to determine the answer, you MUST try 2.  It often screws up your expected answer.  For example:

The product of two prime numbers is always:
(A) odd
(B) even
(C) a prime number
(D) a number with only four factors
(E) It cannot be determined from the information given.

If you forget that 2 is prime, you can try every combination there is of all the other primes, and you will always be left with both (B) and (D) as viable answer choices.  But once you try 2, (B) is eliminated and you are left with (D) as an answer.

Factors and Multiples

What's the difference?

A Factor is a number that goes into another number evenly.  So the factors of 12 are 1, 12, 2, 6, 3, 4.

See how that's written? In school, you're often taught to neatly list factors in order from least to greatest: 1, 2, 3, 4, 6, 12.  Not on this test.
           
-Hint: Always list your factors in their proper pairs, starting with 1 and the number, and then proceeding upwards until you run out of pairs.  Don't put them in proper order until you're looking at the answer choices lest you forget a pair.  Also, don't skip around - start with 1, then see if 2 works, then 3, etc.
           
-Hint: Don't forget 1 and the number itself! These are factors of EVERY number, including primes.  So if a question asks: "How many factors does the number 20 have?" The answer will be "6" NOT "4" And since the other answers are there to trip you up, you can bet "4" will be one of the answer choices.

Prime Factors are the prime numbers that divided into a number evenly.  Remember the factor tree?

 

When you find your prime factors (that is, get to the end of a branch), be sure to circle them as you go so you know you're done and don't forget any.  With larger numbers, it can get messy and confusing on your paper.

Prime factorization is usually what you'll be asked about, but be careful of how it's worded.  If they ask you to list the prime factors of 12, you're answer will be "2, 3." When listing prime factors, you only mention each distinct prime factor once, and you separate them with commas.

If they ask you for the prime factorization of 12, your answer will be "2,3." Prime factorization should ALWAYS multiply back out to the original number, are listed in order from least to greatest using exponents for ones that show up more than once, and are separated by a "times" symbol.

A Multiple is a number you get when you multiply two integers together.  These are the "Multiplication Tables" you had to memorize in grade school.  So some multiples of 8 are 16, 24, 32...

-Hint: Remember, multiples are simply answers you can get when you multiply by an integer - that means, that zero, negatives and the number itself are ALL multiples! So other multiples of 8 include 0, 8, -8, -16, etc.
           
-Tricks to Watch Out For: SAT test takers often get factors and multiples mixed up.  As a result, almost any question that asks you for one will have the other as an answer choice.  For example:

If 2 ≤ w ≤ 100, and w is a multiple of 10 and 35, what is w?
(A) 1
(B) 5
(C) 35
(D) 70
(E) 350

Answer choice (B) is an option strictly for people that forget their definitions of factors and multiples.  As for the others, answer choice (A) is clearly wrong because the question states w must be greater than 2, but is there because 1 is a common factor, and because students don't read the limits carefully.  Answer choice (E) is also there because students don't read the limits carefully.  Without those stated limits on w, (E) would be a fine choice.  (C) is for those that don't consider that multiples of 10 must end in zero.  (D) is the correct answer.

Greatest Common Factor / Least Common Multiple

GCF/LCM
To find the Greatest Common Factor of two numbers, list all the factors of each number and select the one that is the greatest, even if that number is one of the two numbers.

The greatest common factor of 35 and 280 is:
(A) 5
(B) 7
(C) 35
(D) 70
(E) 280

The other way to find the GCF is to use the prime factorization tree.  Break each number down to its prime numbers.  Circle the ones they have in common and multiply those numbers together.

What is the GCF of 425 and 1020?

425 and 1020 have one 5 and one 17 in common.  So the GCF is 85.

The quickest and most accurate way to find the Lowest Common Multiple is also to use the prime factorization tree.  For LCM, however, you're going to keep ONLY ONE of each number that they have in common, multiplying those numbers by all the prime factors left over.  So using 425 and 1020 again, we would keep one 5 and one 17 (cross off the 5 and the 17 listed under 1020 so you don't multiply by them a second time accidentally), and then use all the leftovers.  So the math we need to do would be 5 times 5 times 17 times 2 times 2 times 3.  The LCM of 425 and 1020 is 5100.

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