Patterns in Divisibility Problems

Today we’re going to tackle a couple of tough divisibility problems from GMATPrep. The two problems I’ve chosen share some interesting characteristics. Here’s your first one; set your timer for 2 minutes and go!

If n is a multiple of 5 and n = p ² q , where p and q are prime numbers, which of the following must be a multiple of 25?”
(A)  p ²

(B)  q ²

(C) p q

(D) p²q²

(E) p³q^“

Hmm. So p and q are primes. They could be 2, 3, 5, 7 or so on. It doesn’t say that p and q are different prime numbers, so they could also be the same number. And I’m going to use some theory here: if p²q equals n and n is a multiple of 5, then that 5 must be contained in either p or q. And since those two numbers are primes, either p or q is 5. (Or maybe both are!)

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We invite you to test your GMAT knowledge for a chance to win! Each week, we will post a new Challenge Problem for you to attempt. If you submit the correct answer, you will be entered into that week’s drawing for a free Manhattan GMAT Prep item. Tell your friends to get out their scrap paper and start solving!

Here is this week’s problem:

For positive integers k and n, the k-power remainder of n is defined as r in the following equation:

n = k^w + r, where w is the largest integer such that r is not negative. For instance, the 3-power remainder of 13 is 4, since 13 = 3² + 4. In terms of k and w, what is the largest possible value of r that satisfies the given conditions?

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This week marks the release of our Foundations of GMAT Math Strategy Guide Supplement! This book is one of the first of our Fifth Edition guides to be released, and we are excited for the updates and improvements it represents. Our instructors have been working overtime to make the book more comprehensive and accessible to students on every level, and at 10 chapters and over 500 pages, the new Foundations of GMAT Math is a significant upgrade from the previous edition.

Covering algebra, geometry, number properties, and more, Foundations of GMAT Math was revised to focus not only on foundational knowledge, but on computation in general.  The book now covers all of the basic computation that is tested on the GMAT, teaching students the material through easy-to-follow explanations and cementing understanding with over 700 practice problems.

One of our favorite additions to the book is a series of “If you… Then you…” tables that are designed to make understanding fundamental math principles a snap. We’ve included an example below:

Other improvements include an expanded section on word problems, a clearer problem solving process, and an emphasis on more focused questions.  Additionally, Foundations of GMAT Math now includes more difficult content, including a greater number of questions in the 500-600 range.

With a new cover, a new layout, and an abundance of new content, the Fifth Edition Foundations of GMAT Math is better than ever!  Check it out for yourself by visiting our store.

Also released today was the GMAT Roadmap! Read about it here.

We invite you to test your GMAT knowledge for a chance to win! Each week, we will post a new Challenge Problem for you to attempt. If you submit the correct answer, you will be entered into that week’s drawing for a free Manhattan GMAT Prep item. Tell your friends to get out their scrap paper and start solving!

Here is this week’s problem:

In XYZ Building, a flight of stairs connects each floor to the next, and each flight of stairs is separated from the next flight by a landing. Josie takes twice as long to climb a flight of stairs at a constant rate as she does to cross a landing at another constant rate. If it takes Josie 13.3 minutes to climb 7 flights of stairs and cross the landings between flights, not counting the landings at either end, how long will it take her to climb 10 flights and cross the intervening landings (again not counting landings at either end) at the same rate of travel?

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We invite you to test your GMAT knowledge for a chance to win! Each week, we will post a new Challenge Problem for you to attempt. If you submit the correct answer, you will be entered into that week’s drawing for a free Manhattan GMAT Prep item. Tell your friends to get out their scrap paper and start solving!

Here is this week’s problem:

The ratio, by weight, of the four ingredients A, B, C, and D of a certain mixture is 4:7:8:12. The mixture will be changed so that the ratio of A to C is quadrupled and the ratio of A to D is decreased. The ratio of A to B will be held constant. If B will constitute 20% of the weight of the new mixture, by approximately what percent will the ratio of A to D be decreased?

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We invite you to test your GMAT knowledge for a chance to win! Each week, we will post a new Challenge Problem for you to attempt. If you submit the correct answer, you will be entered into that week’s drawing for a free Manhattan GMAT Prep item. Tell your friends to get out their scrap paper and start solving!

Here is this week’s problem:

The difference between positive two-digit integer A and the smaller two-digit integer B is twice A‘s units digit. What is the hundreds digit of the product of A and B?

(1) The tens digit of A is prime.

(2) Ten is not divisible by the tens digit of A.

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We’ve got another GMATPrep word problem on tap for today, but this one’s in the area of divisibility (number properties). These kinds of problems often include a lot of math vocab; we need to make sure both that we understand the precise words used and concepts being described and that we don’t forget or overlook any of the pieces.

Set your timer for 2 minutes and GO!

If m is a positive odd integer between 2 and 30, then m is divisible by how many different positive prime numbers?
(1) m is not divisible by 3.
(2) m is not divisible by 5.

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In this article, we’re going to tackle a challenging GMATPrep problem solving question from the topic of Percents.  (The GMATPrep software can be downloaded for free at MBA.com)

Let’s start with the problem.

Set your timer for 2 minutes… and… GO!

*Before being simplified, the instructions for computing income tax in country R were to add 2 percent of one’s annual income to the average (arithmetic mean) of 100 units of country R’s currency and 1 percent of one’s annual income. Which of the following represents the simplified formula for computing the income tax, in country R’s currency, for a person in that country whose annual income is I?

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Data sufficiency problems can be a lot of fun because we don’t actually have to solve all the way to the end of the problem. At the same time, data sufficiency problems can be maddening because of the way in which the information is worded. Often, especially on harder questions, the question stem or statements in a data sufficiency problem are worded in such a tricky way that we’re not sure of the significance of the information after we’ve read it.

This lesson is all about how to Rephrase the information in a more useful way. (For those who have taken or are planning to take our class, the Rephrasing lesson occurs during class 1, though I’ve changed the order in which the types are presented in this article.) Read more

This article was written by our very own instructor, Stacey Koprince. If you’re looking for more great resources and advice, check out our free resources section.

I’ve been getting lots of great article ideas from students lately, and this is one of them: how do we make educated guesses on GMAT Quant problems? (I wrote a separate article about Verbal here.)

Note: if you requested a different topic, don’t worry! I keep a list of all requests; I will get to your request eventually. I’m generally choosing the order based upon the number of requests I get from different people about the same topic. Read more