The post Advanced (Non-Calculator!) Factoring on the GMAT appeared first on Magoosh GMAT Blog.

]]>Therefore, if you are aiming to tackle even the 700-800 questions on the Q section, you need to have some crafty mental math tricks up your sleeve. One of the most powerful involves the clever use of a famous algebra formula: the difference of two squares formula. See this post for uses of that formula in general problem solving. Here we will focus on factoring.

In general, factoring a big number can be time-consuming without a calculator. The GMAT might ask a question in which you need to know the factors, or the prime factorization, of a large number. See this post for more on prime factorizations.

First of all, notice how easy it is to square the multiples of 10. If you can square the numbers from 1 to 10, you can square the multiples of 10 from 10 to 100.

Those should all be recognizable as nice round perfect squares.

Now, suppose you are in a situation in which you have to factor, say, 1591, or 3551, or 8099. Notice, all of those are a perfect square less than one of these multiples of ten squared.

In general, the GMAT is not going to put you in a situation in which you have to find the prime factorization of a general four digit number. If this situation does arise, you can bet there’s an enormously simplifying trick available, and factoring via the difference of two squares is an awfully likely candidate for that trick.

Just as the difference of two square can simplify factoring big numbers, it can also simplify factoring decimals. To demonstrate this, I am going to show the solution to a flamboyantly recondite question from the OG13e, viz., PS #199 on p. 180.

199)

(A)

(B)

(C)

(D)

(E)

Notice that , and .

Now, notice that both numerators simplify via difference of two squares formula.

Thus, the two fractions become

The difference:

Answer = D

Here’s a related practice question.

http://gmat.magoosh.com/questions/129

The post Advanced (Non-Calculator!) Factoring on the GMAT appeared first on Magoosh GMAT Blog.

]]>The post Breakdown of GMAT Quant Concepts by Frequency appeared first on Magoosh GMAT Blog.

]]>Here are the samples of Official Material he used to figure out the GMAT Quant breakdown:

1.GMAT Official Guide (12th Edition) Problem Solving Practice Questions (Pg. 152-265)*

2. GMAT Official Guide (12th Edition) Data Sufficiency Practice Questions (Pg. 272-351)*

3. GMATPrep Test #1

4. GMATPrep Test #2

5. Released past exam from GMAC, test code 14, from the 90s/early 2000’s

Below, we have separate rankings for each set of material (#1, #2, #3, #4, #5 ), and at the end, a breakdown by concept type for all of the material combined, as well as a master chart that ranks the frequencies of all the material combined.

Total |
Percentage |
||

Arithmetic | Percents | 21 | 10.71% |

Arithmetic | Properties of Integers | 18 | 9.18% |

Arithmetic | Descriptive Statistics | 12 | 6.12% |

Arithmetic | Fractions | 11 | 5.61% |

Algebra | Linear Equations, Two Unknowns | 11 | 5.61% |

Algebra | Simplifying Algebraic Expressions | 9 | 4.59% |

Arithmetic | Powers & Roots of Numbers | 8 | 4.08% |

Arithmetic | Counting Methods | 7 | 3.57% |

Algebra | Linear Equations, One Unknown | 7 | 3.57% |

Algebra | Functions/Series | 7 | 3.57% |

Geometry | Coordinate Geometry | 7 | 3.57% |

Word Problems | Rate Problems | 7 | 3.57% |

Arithmetic | Ratio & Proportions | 6 | 3.06% |

Arithmetic | Decimals | 5 | 2.55% |

Arithmetic | Discrete Probability | 5 | 2.55% |

Algebra | Exponents | 5 | 2.55% |

Algebra | Inequalities | 5 | 2.55% |

Geometry | Quadrilaterals | 5 | 2.55% |

Geometry | Circles | 5 | 2.55% |

Word Problems | Interest Problems | 4 | 2.04% |

Algebra | Solving Quadratic Equations | 3 | 1.53% |

Geometry | Triangles | 3 | 1.53% |

Geometry | Rectangular Solids & Cylinders | 3 | 1.53% |

Word Problems | Work Problems | 3 | 1.53% |

Word Problems | Mixture Problems | 3 | 1.53% |

Geometry | Intersecting Angles and Lines | 2 | 1.02% |

Word Problems | Profit | 2 | 1.02% |

Word Problems | Sets | 2 | 1.02% |

Word Problems | Measurement Problems | 2 | 1.02% |

Word Problems | Data Interpretation | 2 | 1.02% |

Arithmetic | Real Numbers | 1 | 0.51% |

Arithmetic | Sets | 1 | 0.51% |

Algebra | Solving by Factoring | 1 | 0.51% |

Geometry | Polygons | 1 | 0.51% |

Word Problems | Discount | 1 | 0.51% |

Word Problems | Geometry Problems | 1 | 0.51% |

Algebra | Equations | 0 | 0.00% |

Algebra | Absolute Value | 0 | 0.00% |

Geometry | Lines | 0 | 0.00% |

Geometry | Perpendicular Lines | 0 | 0.00% |

Geometry | Parallel Lines | 0 | 0.00% |

Total |
Percentage |
||

Arithmetic | Properties of Integers | 30 | 19.23% |

Arithmetic | Descriptive Statistics | 13 | 8.33% |

Arithmetic | Percents | 11 | 7.05% |

Algebra | Linear Equations, Two Unknowns | 11 | 7.05% |

Word Problems | Rate Problems | 9 | 5.77% |

Algebra | Inequalities | 8 | 5.13% |

Arithmetic | Sets | 7 | 4.49% |

Arithmetic | Ratio & Proportions | 6 | 3.85% |

Geometry | Triangles | 6 | 3.85% |

Geometry | Circles | 6 | 3.85% |

Arithmetic | Decimals | 5 | 3.21% |

Geometry | Coordinate Geometry | 5 | 3.21% |

Arithmetic | Fractions | 4 | 2.56% |

Algebra | Linear Equations, One Unknown | 4 | 2.56% |

Algebra | Exponents | 4 | 2.56% |

Geometry | Rectangular Solids & Cylinders | 4 | 2.56% |

Arithmetic | Discrete Probability | 3 | 1.92% |

Algebra | Functions/Series | 3 | 1.92% |

Word Problems | Interest Problems | 3 | 1.92% |

Geometry | Lines | 2 | 1.28% |

Word Problems | Work Problems | 2 | 1.28% |

Word Problems | Discount | 2 | 1.28% |

Word Problems | Profit | 2 | 1.28% |

Arithmetic | Real Numbers | 1 | 0.64% |

Arithmetic | Counting Methods | 1 | 0.64% |

Algebra | Simplifying Algebraic Expressions | 1 | 0.64% |

Algebra | Absolute Value | 1 | 0.64% |

Geometry | Quadrilaterals | 1 | 0.64% |

Word Problems | Measurement Problems | 1 | 0.64% |

Arithmetic | Powers & Roots of Numbers | 0 | 0.00% |

Algebra | Equations | 0 | 0.00% |

Algebra | Solving by Factoring | 0 | 0.00% |

Algebra | Solving Quadratic Equations | 0 | 0.00% |

Geometry | Intersecting Angles and Lines | 0 | 0.00% |

Geometry | Perpendicular Lines | 0 | 0.00% |

Geometry | Parallel Lines | 0 | 0.00% |

Geometry | Polygons | 0 | 0.00% |

Word Problems | Mixture Problems | 0 | 0.00% |

Word Problems | Sets | 0 | 0.00% |

Word Problems | Geometry Problems | 0 | 0.00% |

Word Problems | Data Interpretation | 0 | 0.00% |

Total |
Percentage |
||

Arithmetic | Properties of Integers | 6 | 17.14% |

Arithmetic | Ratio & Proportions | 4 | 11.43% |

Arithmetic | Descriptive Statistics | 4 | 11.43% |

Algebra | Exponents | 3 | 8.57% |

Algebra | Functions/Series | 3 | 8.57% |

Arithmetic | Percents | 2 | 5.71% |

Arithmetic | Powers & Roots of Numbers | 2 | 5.71% |

Word Problems | Interest Problems | 2 | 5.71% |

Arithmetic | Fractions | 1 | 2.86% |

Arithmetic | Counting Methods | 1 | 2.86% |

Algebra | Equations | 1 | 2.86% |

Geometry | Intersecting Angles and Lines | 1 | 2.86% |

Geometry | Triangles | 1 | 2.86% |

Geometry | Circles | 1 | 2.86% |

Word Problems | Work Problems | 1 | 2.86% |

Word Problems | Sets | 1 | 2.86% |

Word Problems | Measurement Problems | 1 | 2.86% |

Arithmetic | Decimals | 0 | 0.00% |

Arithmetic | Real Numbers | 0 | 0.00% |

Arithmetic | Sets | 0 | 0.00% |

Arithmetic | Discrete Probability | 0 | 0.00% |

Algebra | Simplifying Algebraic Expressions | 0 | 0.00% |

Algebra | Linear Equations, One Unknown | 0 | 0.00% |

Algebra | Linear Equations, Two Unknowns | 0 | 0.00% |

Algebra | Solving by Factoring | 0 | 0.00% |

Algebra | Solving Quadratic Equations | 0 | 0.00% |

Algebra | Inequalities | 0 | 0.00% |

Algebra | Absolute Value | 0 | 0.00% |

Geometry | Lines | 0 | 0.00% |

Geometry | Perpendicular Lines | 0 | 0.00% |

Geometry | Parallel Lines | 0 | 0.00% |

Geometry | Polygons | 0 | 0.00% |

Geometry | Quadrilaterals | 0 | 0.00% |

Geometry | Rectangular Solids & Cylinders | 0 | 0.00% |

Geometry | Coordinate Geometry | 0 | 0.00% |

Word Problems | Rate Problems | 0 | 0.00% |

Word Problems | Mixture Problems | 0 | 0.00% |

Word Problems | Discount | 0 | 0.00% |

Word Problems | Profit | 0 | 0.00% |

Word Problems | Geometry Problems | 0 | 0.00% |

Word Problems | Data Interpretation | 0 | 0.00% |

Total |
Percentage |
||

Arithmetic | Descriptive Statistics | 6 | 18.75% |

Arithmetic | Properties of Integers | 4 | 12.50% |

Arithmetic | Ratio & Proportions | 2 | 6.25% |

Arithmetic | Percents | 2 | 6.25% |

Algebra | Simplifying Algebraic Expressions | 2 | 6.25% |

Algebra | Exponents | 2 | 6.25% |

Geometry | Coordinate Geometry | 2 | 6.25% |

Arithmetic | Fractions | 1 | 3.13% |

Arithmetic | Powers & Roots of Numbers | 1 | 3.13% |

Arithmetic | Counting Methods | 1 | 3.13% |

Algebra | Equations | 1 | 3.13% |

Algebra | Linear Equations, Two Unknowns | 1 | 3.13% |

Algebra | Inequalities | 1 | 3.13% |

Algebra | Functions/Series | 1 | 3.13% |

Geometry | Triangles | 1 | 3.13% |

Geometry | Circles | 1 | 3.13% |

Word Problems | Rate Problems | 1 | 3.13% |

Word Problems | Discount | 1 | 3.13% |

Word Problems | Measurement Problems | 1 | 3.13% |

Arithmetic | Decimals | 0 | 0.00% |

Arithmetic | Real Numbers | 0 | 0.00% |

Arithmetic | Sets | 0 | 0.00% |

Arithmetic | Discrete Probability | 0 | 0.00% |

Algebra | Linear Equations, One Unknown | 0 | 0.00% |

Algebra | Solving by Factoring | 0 | 0.00% |

Algebra | Solving Quadratic Equations | 0 | 0.00% |

Algebra | Absolute Value | 0 | 0.00% |

Geometry | Lines | 0 | 0.00% |

Geometry | Intersecting Angles and Lines | 0 | 0.00% |

Geometry | Perpendicular Lines | 0 | 0.00% |

Geometry | Parallel Lines | 0 | 0.00% |

Geometry | Polygons | 0 | 0.00% |

Geometry | Quadrilaterals | 0 | 0.00% |

Geometry | Rectangular Solids & Cylinders | 0 | 0.00% |

Word Problems | Work Problems | 0 | 0.00% |

Word Problems | Mixture Problems | 0 | 0.00% |

Word Problems | Interest Problems | 0 | 0.00% |

Word Problems | Profit | 0 | 0.00% |

Word Problems | Sets | 0 | 0.00% |

Word Problems | Geometry Problems | 0 | 0.00% |

Word Problems | Data Interpretation | 0 | 0.00% |

Total |
Percentage |
||

Arithmetic | Properties of Integers | 7 | 13.73% |

Arithmetic | Ratio & Proportions | 5 | 9.80% |

Arithmetic | Percents | 5 | 9.80% |

Arithmetic | Descriptive Statistics | 4 | 7.84% |

Word Problems | Sets | 4 | 7.84% |

Arithmetic | Powers & Roots of Numbers | 3 | 5.88% |

Algebra | Linear Equations, Two Unknowns | 3 | 5.88% |

Word Problems | Rate Problems | 3 | 5.88% |

Arithmetic | Fractions | 2 | 3.92% |

Arithmetic | Counting Methods | 2 | 3.92% |

Algebra | Exponents | 2 | 3.92% |

Algebra | Inequalities | 2 | 3.92% |

Geometry | Triangles | 2 | 3.92% |

Algebra | Simplifying Algebraic Expressions | 1 | 1.96% |

Algebra | Equations | 1 | 1.96% |

Algebra | Functions/Series | 1 | 1.96% |

Geometry | Quadrilaterals | 1 | 1.96% |

Geometry | Circles | 1 | 1.96% |

Geometry | Rectangular Solids & Cylinders | 1 | 1.96% |

Geometry | Coordinate Geometry | 1 | 1.96% |

Arithmetic | Decimals | 0 | 0.00% |

Arithmetic | Real Numbers | 0 | 0.00% |

Arithmetic | Sets | 0 | 0.00% |

Arithmetic | Discrete Probability | 0 | 0.00% |

Algebra | Linear Equations, One Unknown | 0 | 0.00% |

Algebra | Solving by Factoring | 0 | 0.00% |

Algebra | Solving Quadratic Equations | 0 | 0.00% |

Algebra | Absolute Value | 0 | 0.00% |

Geometry | Lines | 0 | 0.00% |

Geometry | Intersecting Angles and Lines | 0 | 0.00% |

Geometry | Perpendicular Lines | 0 | 0.00% |

Geometry | Parallel Lines | 0 | 0.00% |

Geometry | Polygons | 0 | 0.00% |

Word Problems | Work Problems | 0 | 0.00% |

Word Problems | Mixture Problems | 0 | 0.00% |

Word Problems | Interest Problems | 0 | 0.00% |

Word Problems | Discount | 0 | 0.00% |

Word Problems | Profit | 0 | 0.00% |

Word Problems | Geometry Problems | 0 | 0.00% |

Word Problems | Measurement Problems | 0 | 0.00% |

Word Problems | Data Interpretation | 0 | 0.00% |

Total |
Percentage |
||

Arithmetic |
Properties of Integers | 65 | 13.83% |

Fractions | 19 | 4.04% | |

Decimals | 10 | 2.13% | |

Real Numbers | 2 | 0.43% | |

Ratio & Proportions | 23 | 4.89% | |

Percents | 41 | 8.72% | |

Powers & Roots of Numbers | 14 | 2.98% | |

Descriptive Statistics | 39 | 8.30% | |

Sets | 8 | 1.70% | |

Counting Methods | 12 | 2.55% | |

Discrete Probability | 8 | 1.70% | |

Algebra |
Simplifying Algebraic Expressions | 13 | 2.77% |

Equations | 3 | 0.64% | |

Linear Equations, One Unknown | 11 | 2.34% | |

Linear Equations, Two Unknowns | 26 | 5.53% | |

Solving by Factoring | 1 | 0.21% | |

Solving Quadratic Equations | 3 | 0.64% | |

Exponents | 16 | 3.40% | |

Inequalities | 16 | 3.40% | |

Absolute Value | 1 | 0.21% | |

Functions/Series | 15 | 3.19% | |

Geometry |
Lines | 2 | 0.43% |

Intersecting Angles and Lines | 3 | 0.64% | |

Perpendicular Lines | 0 | 0.00% | |

Parallel Lines | 0 | 0.00% | |

Polygons | 1 | 0.21% | |

Triangles | 13 | 2.77% | |

Quadrilaterals | 7 | 1.49% | |

Circles | 14 | 2.98% | |

Rectangular Solids & Cylinders | 8 | 1.70% | |

Coordinate Geometry | 15 | 3.19% | |

Word Problems |
Rate Problems | 20 | 4.26% |

Work Problems | 6 | 1.28% | |

Mixture Problems | 3 | 0.64% | |

Interest Problems | 9 | 1.91% | |

Discount | 4 | 0.85% | |

Profit | 4 | 0.85% | |

Sets | 7 | 1.49% | |

Geometry Problems | 1 | 0.21% | |

Measurement Problems | 5 | 1.06% | |

Data Interpretation | 2 | 0.43% | |

Total |
470 | 100.00% |

Total |
Percentage |
||

Arithmetic | Properties of Integers | 65 | 13.83% |

Arithmetic | Percents | 41 | 8.72% |

Arithmetic | Descriptive Statistics | 39 | 8.30% |

Algebra | Linear Equations, Two Unknowns | 26 | 5.53% |

Arithmetic | Ratio & Proportions | 23 | 4.89% |

Word Problems | Rate Problems | 20 | 4.26% |

Arithmetic | Fractions | 19 | 4.04% |

Algebra | Exponents | 16 | 3.40% |

Algebra | Inequalities | 16 | 3.40% |

Algebra | Functions/Series | 15 | 3.19% |

Geometry | Coordinate Geometry | 15 | 3.19% |

Arithmetic | Powers & Roots of Numbers | 14 | 2.98% |

Geometry | Circles | 14 | 2.98% |

Algebra | Simplifying Algebraic Expressions | 13 | 2.77% |

Geometry | Triangles | 13 | 2.77% |

Arithmetic | Counting Methods | 12 | 2.55% |

Algebra | Linear Equations, One Unknown | 11 | 2.34% |

Arithmetic | Decimals | 10 | 2.13% |

Word Problems | Interest Problems | 9 | 1.91% |

Arithmetic | Sets | 8 | 1.70% |

Arithmetic | Discrete Probability | 8 | 1.70% |

Geometry | Rectangular Solids & Cylinders | 8 | 1.70% |

Geometry | Quadrilaterals | 7 | 1.49% |

Word Problems | Sets | 7 | 1.49% |

Word Problems | Work Problems | 6 | 1.28% |

Word Problems | Measurement Problems | 5 | 1.06% |

Word Problems | Discount | 4 | 0.85% |

Word Problems | Profit | 4 | 0.85% |

Algebra | Equations | 3 | 0.64% |

Algebra | Solving Quadratic Equations | 3 | 0.64% |

Geometry | Intersecting Angles and Lines | 3 | 0.64% |

Word Problems | Mixture Problems | 3 | 0.64% |

Arithmetic | Real Numbers | 2 | 0.43% |

Geometry | Lines | 2 | 0.43% |

Word Problems | Data Interpretation | 2 | 0.43% |

Algebra | Solving by Factoring | 1 | 0.21% |

Algebra | Absolute Value | 1 | 0.21% |

Geometry | Polygons | 1 | 0.21% |

Word Problems | Geometry Problems | 1 | 0.21% |

Geometry | Perpendicular Lines | 0 | 0.00% |

Geometry | Parallel Lines | 0 | 0.00% |

The list of concepts tested on the quantitative section is from GMAC, which you can see on page 107 of the Official Guide (either 12th or 13th edition). It isn’t perfect– “Integer Properties” is a wide area of knowledge, whereas something like “Circles” is very specific.

- Based on the master chart alone, Arithmetic is the clear “winner”, while Word Problems and most Geometry question are ranked much lower.
- Though the rankings vary slightly from chart to chart, there are no extreme outliers in terms of the sets of data– even the old released exam is quite consistent with all of the other exams.

NO. For the sake of simplicity and accuracy in reporting absolute frequency, we’ve only assigned each question to one concept. This means that even though GMAC lists “Perpendicular lines” as a topic tested on the GMAT, and we have 0 questions marked as pertaining to that topic, that certainly doesn’t mean the idea of perpendicular lines did not come up at all on all of the exams. It certainly appeared, but often in questions that were better categorized, overall, as “Coordinate Geometry”, or “Intersecting Angles and Lines”.

We hope this serves as a guideline for the relative frequency of math topics tested on the GMAT to help you decide how to focus your time! In Magoosh practice, you can set up customized practice sessions to focus on specific concepts, as well as review your performance on individual concepts to identify your weak spots using our Review tool.

Let us know whether you find this type of breakdown helpful, and whether you have any questions about any of the information above! 🙂

The post Breakdown of GMAT Quant Concepts by Frequency appeared first on Magoosh GMAT Blog.

]]>The post GMAT Divisibility Rules and Shortcuts appeared first on Magoosh GMAT Blog.

]]>Fact: On the GMAT, you do not get a calculator.

Actually, on the new IR questions coming this summer, there will be an onscreen calculator, but for the ordinary Quantitative questions, for the foreseeable future, you will be doing those without a calculator of any sort.

Consider the following question:

S = {71, 73, 75, 77, 79, 81, 83, 85, 87, 89}

How many numbers in S are prime?

This, with little variation, could be a Problem Solving question, or it could play a role in some aspect of a Data Sufficiency question. How to determine this without a calculator?

First of all, it’s probably obvious that all even numbers are divisible by 2. Furthermore, 2 is the only even prime number: all other positive even integers are divisible by 2, so all other prime numbers are odd.

A number that ends with a 5 or a 0 is divisible by 5. A two digit number in which the first and last digits are the same is divisible by 11.

With 4, there’s an interesting pattern. Think of the one-digit numbers that are multiples of 4: those are 0, 4, and 8. If the last digit is 0, 4, or 8, and the tens digit is *even*, then the number is divisible by 4. If the last digit is even but not one of those — viz. 2 or 6, the one-digit even numbers that are *not* multiples of 4 – and the tens digit is *odd*, then the number is divisible by 4.

The rule for 3 is a bit different from the rules for other numbers. You simply add up the digits. If the sum of the digits is divisible by 3, than the original number is divisible by 3. For example, consider the number 285 —- 2 + 8 + 5 = 15, which is divisible by 3, so that means 285 must be divisible by 3. Consider the number 2012 —- 2 + 0 + 1 + 2 = 5, which is not divisible by 3, so that means 2012 is not divisible by 3.

That pattern also works with 9 — if the sum of the digits is divisible by 9, then the original number is divisible by 9. For example, the next year that will be a multiple of 9 is 2016, because 2 + 0 + 1 + 6 = 9.

The GMAT occasionally will have questions that require you to figure out whether larger two-digit numbers are prime. It will not ask you to figure out whether numbers greater than 100 are prime. For numbers less than 100, two-digit numbers, it’s enough to check whether it divisible by any single digit prime number {2, 3, 5, 7}. Any two-digit number not divisible by one of those four must be prime.

Back to our set above:

S = {71, 73, 75, 77, 79, 81, 83, 85, 87, 89}

All odds, so no multiples of 2.The numbers 75 and 85 are obviously divisible by 5, so those are not prime.

S = {71, 73, __, 77, 79, 81, 83, __, 87, 89}

The number 77 = 7*11. Once we have that multiple of 7, we add and subtract 7 to find other multiples of 7: { . . . . 70, 77, 84, 91, . . . } We see that 77 is the only multiple of 7 on this list.

S = {71, 73, __, __, 79, 81, 83, __, 87, 89}

Now we need a multiple of 3. We notice that, with 81, 8+1=9, which is divisible by 3, so 81 must be divisible by 3. We add and subtract 3 repeatedly to get other multiples of three: { . . . 69, 72, 75, 78, 81, 84, 87, 90, . . .}. Just to double-check 87, we see 8+7=15, which is divisible by 3, so of course 87 is divisible by 3. We can eliminate

S = {71, 73, __, __, 79, __, 83, __, __, 89}

The remaining five numbers must be prime. Indeed, each one is a prime number.

Here’s a free GMAT Problem Solving question on which you can practice these skills: http://gmat.magoosh.com/questions/850

That question will be followed by a full video explanation when you submit your answer. Good luck!

The post GMAT Divisibility Rules and Shortcuts appeared first on Magoosh GMAT Blog.

]]>The post GMAT Math: Factors appeared first on Magoosh GMAT Blog.

]]>“If n is the smallest integer such that 432 times n is the square of an integer, what is the value of y?”

“How many prime numbers are factors of 33150?”

If questions like these make you cringe, I’d like to convince you that only a few easy-to-understand concepts stand between you and doing these flawlessly.

This is probably review, but just for a refresher: a prime number is any positive integer that is divisible by only 1 and itself. In other words, a prime number has only two factors: itself and 1. Numbers that have more than two factors are called composite. (By mathematical convention, 1 is the only positive integer considered neither prime nor composite.) The first few prime numbers are:

2 3 5 7 11 13 17 19 23 29

In preparation for the GMAT, it would be good to be familiar with this list. If you verify for yourself why each number from 2 to 30 is prime or composite, it will help you remember this list.

Occasionally, the GMAT will expect you know whether a larger two-digit number, like 67, is prime. Of course, if the number is even, it’s not prime. If it ends in a digit of 5, it’s divisible by five. For divisibility by 3, a good trick to know: if the sum of the digits is divisible by three, then the number is divisible by three. Here 6 + 7 = 13, not divisible by three, so 67 is not divisible by three.

To see whether a number less than 100 is prime, all we have to do is see whether it is divisible by one of the single digit prime numbers: 2, 3, 5, or 7. We’ve already checked 2, 3, and 5. The number 67 is not divisible by 7: 7 goes evenly into 63 and 70, not 67. That’s enough checking to establish irrevocably that 67 is prime.

Every positive integer greater than 1 can be written in a unique way as a product of prime numbers; this is called its **prime factorization**. The prime factorization is analogous to the DNA of the number, the unique blueprint by which to construct the number. In other words, when you calculate the prime factorization of a number, you have some powerful information at your disposal.

How does one calculate the prime factorization of a number? In grade school, you may remember making “factor trees”: that’s the idea. To find the prime factorization of, for example, 48, we simply choose any two factors — say 6 and 8 — and then choose factors of those number, and then of those numbers, until we are left with nothing but primes.

Typically, once we are done, we sort the prime factors in numerical order:

Once we have the prime factorization, what can we do with it? See the next two items.

Suppose the GMAT asks: how many factors does 1440 have? It would be quite tedious to count them all, but there’s a fast trick once you have the prime factorization. First of all, the prime factorization of 1440 is

Each prime factor has an exponent (the exponent of 5 is 1).

To find the total number of factors:

a) Find the list of exponents in the prime factorization — here {5, 2, 1}

b) Add one to each number on the list — here {6, 3, 2}

c) Multiply those together —

The number 1440 has thirty-six factors, including 1 and itself.

Suppose the GMAT asked the number of odd factors of 1440. We know that odd factors cannot contain any factor of 2 at all, so basically we repeat that procedure with all the factors except the factors of 2. Here: {2, 1} –> {3, 2} –> . The number 1440 has 6 odd factors, including 1. Just for verification, the odd factors of 1440 are

{1, 3, 5, 9, 15, and 45}

This also means it has 36 – 6 = 30 even factors.

GCF = greatest common factor

LCM = least common multiple.

(Note: LCM and LCD are the same thing: a least common denominator, LCD, of two number is always their LCM.)

Suppose a GMAT Math question involves finding, say, the LCM (or LCD) of 30 and 48. There’s a very straightforward procedure to find the LCM.

- Find the prime factorizations of the two numbers: 30 = 2*3*5 and 48 = 2*2*2*2*3
- Find the factors they have in common – the product of these is the GCF. Here, the GCF = 2*3 = 6
- Express each number as the GCF*(other stuff): 30 = 6*5 and 48 = 6*8
- The LCM = GCF*(other stuff from first number)*(other stuff from the second number): LCM = 6*5*8 = 240

Let’s do one more, just for practice. Suppose, on a GMAT math problem, we need to find the LCM/LCD of 28 and 180

Step (a): 28 = 2*2*7, 180 = 2*2*3*3*5

Step (b): 28 = **2*2***7, 180 = **2*2***3*3*5; GCF = 2*2 = 4

Step (c) 28 = 4*7, 180 = 4*45

Step (d) LCM = 4*7*45 = 1260

1) The number of boxes in a warehouse can be divided evenly into 6 equal shipments by boat or 27 equal shipments by truck. What is the smallest number of boxes that could be in the warehouse?

(A) 27

(B) 33

(C) 54

(D) 81

(E) 162

2) How many odd factors does 210 have?

(A) 3

(B) 4

(C) 5

(D) 6

(E) 8

3) If n is the smallest integer such that 432 times n is the square of an integer, what is the value of n?

(A) 2

(B) 3

(C) 6

(D) 12

(E) 24

4) How many distinct prime numbers are factors of 33150?

(A) Four

(B) Five

(C) Six

(D) Seven

(E) Eight

5) If n is a positive integer, then n(n + 1)(n – 1) is

(A) even only when n is even

(B) odd only when n is even

(C) odd only when n is odd

(D) always divisible by 3

(E) always one less than a prime number

1) C

2) E

3) B

4) B

5) D

1) This tells us that the number of boxes is evenly divisible by both 6 and 27; in other words, it’s a common multiple of 6 and 27. The question says: what’s the smallest value it could have? In other words, what’s the LCM of 6 and 27? (This question is one example of a real-world set-up where the question is actually asking for the LCM.)

Step (a): 6 = 2*3 27 = 3*3*3

Step (b): 6 = 2*3 27 = 3*3*3 GCF = 3

Step (c): 6 = 3*2 27 = 3*9

Step (d) LCM = 3*2*9 = 54

Thus, 54 is the LCM of 6 and 27.

**Answer: C.**

2) Start with the prime factorization: 210 = 2*3*5*7

For odd factors, we put aside the factor of two, and look at the other prime factors.

set of exponents = {1, 1, 1}

plus 1 to each = {2, 2, 2}

product = 2*2*2 = 8

Therefore, there are 8 odd factors of 210. In case you are curious, they are {1, 3, 5, 7, 15, 21, 35, and 105}

**Answer: E.**

3) The prime factorization of a square has to have even powers of all its prime factors. If the original number has a factor, say of 7, then when it’s squared, the square will have a factor of 7^2. Another way to say that is: any positive integer all of whose prime factors have even powers must be a perfect square of some other integer. Look at the prime factorization of 432

432 = (2^4)*(3^3)

The factor of 2 already has an even power —- that’s all set. The factor of 3 currently has an odd power. If n = 3, then 432*n would have an even power of 2 and an even power of 3; therefore, it would be a perfect square. Thus, n = 3 is a choice that makes 432*n a perfect square.

**Answer: B.**

4) Start with the prime factorization:

33150 = 50*663 = (2*5*5)*3*221 = (2)*(3)*(5^2)*(13)*(17)

There are five distinct prime factors, {2, 3, 5, 13, and 17}

**Answer: B.**

5) Notice that (n – 1) and n and (n + 1) are three consecutive integers. This question is about the product of three consecutive integers.

If n is even, then this product will be (odd)*(even)*(odd) = even

If n is odd, this this product will be (even)*(odd)*(even) = even

No matter what, the product is even. Therefore, answers (A) & (B) & (C) are all out.

Let’s look at a couple examples, to get a feel for this

3*4*5 = 60

4*5*6 = 120

5*6*7 = 210

6*7*8 = 336

7*8*9 = 504

Notice that one of the three numbers always has to be a multiple of 3: when you take any three consecutive integers, one of them is always a multiple of 3. Therefore the product will always be divisible by 3.

Therefore, **Answer: D.**

BTW, for answer choice E of that question, you will notice that for some trios of positives integers, adding one to the product does result in a prime, but for others, it doesn’t.

3*4*5 + 1 = 61 = prime

4*5*6 + 1 = 121 = 11^2 (not prime)

5*6*7 + 1 = 211 = prime

6*7*8 + 1 = 337 = prime

7*8*9 + 1 = 505 = 5*101 (not prime)

This is a mathematical idea far far more advanced than anything on the GMAT, but it is mathematically impossible to create an easy rule or formula that will always result in prime numbers. The prime numbers follow an astonishingly complicated pattern, which is the subject of the single hardest unanswered question in modern mathematics: the Riemann Hypothesis. Fascinating stuff for leisure reading, but absolutely 100% not needed for the GMAT :).

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