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.

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