Multiples

Arithmetic: Multiples

Read the following SAT test question and then select the correct answer.

Approach each problem the same way so that you feel confident about your ability to solve it.  Start by reading the question carefully and identifying your bottom line.  Then assess your options and choose the most efficient method to attack the problem.  When you have an answer, loop back to verify that the answer addresses the bottom line.

Add 8x to 2x and then subtract 5 from the sum. If x is a positive integer, the result must be an integer multiple of

Bottom Line:  multiple of = ?

Assess your Options:  You have to write an equation for this problem, but after doing so you can use logic or the strategy of plugging in numbers to find possible answers to the equation.  Both methods are quick and will result in the correct answer.

Attack the Problem:  Your first step is to translate all the words you are given into an equation. If you add 8x to 2x, you get 8x + 2x.  Then subtract 5.  You should have:

8x + 2x – 5

Always simplify as much as possible before moving to the next step.  Here, you can combine like terms.

10x – 5

Now go back to the other information that you are given.  The variable x must be a positive integer.  Plug in the smallest possible value for x, and you will get the smallest possible result of this equation.  Plug in x = 1.

10(1) – 5 = 5

Now, multiples will always get larger, so there are other possible answers to this equation.  However, this is the smallest answer and you are looking for what the result “must” be an integer multiple of.  Multiples are simply the product of a number and an integer.  5 is a prime number, so the only thing that the answer must be a multiple of is 5.

(If you want to make sure you are on the right track, plug in x = 2.  The answer is 15.  15 is still a multiple of 5.  Any positive number that you plug in will still be a multiple of 5 because when you subtract 5 from a multiple of 10, you will always get a number ending in a 5.)

Loop Back:  You found that the answer must be a multiple of 5.  Look down at your answer choices.

(A) 2
(B) 5
(C) 8
(D) 10
(E) 15

The correct answer is (B).

On sat.collegeboard.org, 68% of the responses were correct.

For more help with SAT math, visit www.myknowsys.com!

Sets

Arithmetic: Sets

Read the following SAT test question and then select the correct answer.

Approach each math question on the SAT the same way.  Read the question carefully to be sure you take into account all of the information as you solve it, and be sure to identify and note the bottom line.  Assess your options for solving the problem, and then choose the most efficient method to attack the problem.  Never forget to loop back and make sure that your final answer solves for the bottom line, the question that you were asked.

If S is the set of positive integers that are multiples of 7, and if T is the set of positive integers that are multiples of 13, how many integers are in the intersection of S and T?

Bottom Line: # of intersections = ?

Assess your Options:  When you have a question that asks about number properties, ignore your answer choices!  If you look down and see a 0, you could think to yourself that both 7 and 13 are prime, so they have nothing in common.  Are you looking for factors?  No!  You are looking for multiples.  Think through all of the information that you are given before looking at the answer choices.

Attack the Problem:  A set is just a collection of data.  You are given two different sets and asked to find the intersections, the data that the two have in common.  The only restriction on both sets is that all of the numbers must be positive.

Now think about what multiples are.  Multiples are the product of a number and an integer.  So Set S contains 7, 14, 21, 28… and continues in this manner into infinity.  Set T contains 13, 26, 39, 52… and continues in this manner into infinity.

If you keep listing numbers in each set, it will take you forever to find the answer to this problem.  Instead, think logically about where you know you must have multiples that match.  For example, if you multiply 7 times 13, you will find a number that belongs in both sets.  If you multiply 14 times 13, you will find another intersection.  Notice that you can keep doing this because you will never reach infinity.  The answer to this problem is that there are an infinite number of intersections between S and T.

Loop Back:  You found your bottom line, so look down and see which answer choice it matches.

(A) None
(B) One
(C) Seven
(D) Thirteen
(E) More than thirteen

The correct answer is (E).

On sat.collegeboard.org, 40% of the responses were correct.

For more help with SAT math, visit www.myknowsys.com!

Sets

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Mathematics: Standard Multiple Choice

Read the following SAT test question and then select your answer.

Always follow the Knowsys Method to save time and energy on math questions: read carefully, identify the bottom line, assess your options, attack the problem, and loop back to double-check your answer. This will help you find the correct answers more quickly.

If S is the set of positive integers that are multiples of 7, and if T is the set of positive integers that are multiples of 13, how many integers are in the intersection of S and T?

First, read carefully. Notice that S includes all multiples of 7 and T includes all multiples of 13. Next, identify the bottom line and note it at the top of your scratch work.

Intersection of S and T = ?

You are looking for the intersection, so you need only the numbers that are in both set S and set T. Since neither set has an upper limit, they both have an infinite number of members; therefore, their intersection also has an infinite number of members. Look at the answer choices.

A) None

B) One

C) Seven

D) Thirteen

E) More than thirteen

The answer is E.