Math and the Integrated Reasoning Section


Before we get started talking about the Quantitative section of the GMAT, let’s take a moment to talk about the Integrated Reasoning section. The Integrated Reasoning section tests a blend of math and critical reasoning (verbal) skills. However, some questions test just math skills. The good news is that those math skills are the same math skills that the GMAT has been testing for years. So, while we’ll be mostly discussing the Quantitative section in this and the following chapters, remember that you’ll also use the math reviewed in these chapters to answer some of the Integrated Reasoning questions.



在我們開始討論GMAT的“定量”部分之前,讓我們花一點時間討論“綜合推理”部分。 綜合推理部分測試數學和批判性推理(語言)技能的融合。 但是,有些問題只是測試數學技能。 好消息是這些數學技能與GMAT多年來測試的數學技能相同。 因此,儘管我們將在本章及以下各章中主要討論“定量”部分,但請記住,您還將使用這些章中回顧的數學來回答一些綜合推理問題。

What’s Covered in the Quantitative Section

The 31 math questions on the GMAT come in two different formats. About half of the questions are regular Problem Solving questions of the type you’re familiar with from countless other standardized tests, such as the SAT. The other half of the questions, mixed in among the regular Problem Solving questions, are of a type unique to the GMAT: they’re called Data Sufficiency questions, and they ask you to determine whether you can answer a math question based on two pieces of information. We’ve devoted two entire chapters to Data Sufficiency.


But whether the question falls into the category of Problem Solving or Data Sufficiency, GMAT questions test your general knowledge of three subjects:


GMAT上的31個數學問題有兩種不同的格式。 大約一半的問題是您通過無數其他標準化測試(例如SAT)所熟悉的常規問題解決問題。 問題的另一半,混合在常規問題解決問題中,是GMAT特有的類型:它們稱為數據充足性問題,它們要求您根據兩個問題來確定是否可以回答數學問題 信息。 我們在整整兩章中專門介紹了數據充分性。



1. Arithmetic

2. Basic algebra

3. Basic geometry




What Isn’t Covered in the Quantitative Section

The good news is that you won’t need to know calculus, trigonometry, or any complicated geometry. The bad news is that the specialized, business-type math you’re probably good at isn’t tested, either. There will be no questions on computing the profit on three ticks of a particular bond sale, no questions about amortizing a loan, no need to calculate the bottom line of a small business.


好消息是,您無需了解微積分,三角函數或任何復雜的幾何圖形。 壞消息是,您可能擅長的專業商務類數學也沒有經過測試。 毫無疑問,可以計算特定債券出售的三個滴答滴答的利潤,也沒有關於攤銷貸款的問題,也不需要計算小企業的底線。

Ancient History

For the most part, what you’ll find on the GMAT is a kind of math that you haven’t had to think about in years: junior high school and high school math. Because most people who apply to business school have been out of college for several years, high school math may seem a bit like ancient history. In the next few chapters, we’ll give you a fast review of the important concepts, and we’ll show you some powerful techniques for cracking the system.


在大多數情況下,您會在GMAT上找到的是一種多年來無需考慮的數學:初中和高中數學。 因為大多數申請商學院的人都已經大學畢業了幾年,所以高中數學似乎有點像古老的歷史。 在接下來的幾章中,我們將為您快速回顧重要概念,並向您展示一些強大的技術來破解系統。

The Princeton Review Approach

Because it’s probably been a long time since you’ve needed to reduce fractions or remember how many degrees there are in a quadrilateral, the first thing to do is review the information tested on the GMAT by going through our math review. Along the way, you’ll learn some valuable test-taking skills that will allow you to take advantage of some of the inherent weaknesses of standardized testing.


When you’ve finished the math review, you should read our chapter on Data Sufficiency and then take our diagnostic math test. Based on your approximate score on our diagnostic, you can then practice working through problems at, or just above, your scoring range. By becoming familiar with the general level of difficulty of these problems and the number of steps required to solve them, you can increase your score on the real GMAT.


因為您可能需要很長時間才能減少分數或記住四邊形有多少個度數,所以第一件事就是通過我們的數學審查來複習在GMAT上測試的信息。 在此過程中,您將學習一些有價值的考試技巧,這些技巧將使您能夠利用標準化考試的某些固有弱點。


完成數學審查後,您應該閱讀我們的數據充分性一章,然後進行診斷數學測試。 根據您在我們的診斷程序中的近似分數,您可以練習解決分數範圍或以上的問題。 通過熟悉這些問題的一般難度以及解決這些問題所需的步驟數,您可以提高實際GMAT的分數。

Extra Help

Although we can show you which mathematical principles are most important for the GMAT, this book cannot take the place of a basic foundation in math. We find that most people, even if they don’t remember much of high school math, pick it up again quickly. Our drills and examples will refresh your memory if you’ve gotten rusty, but if you have serious difficulties with the following chapters, you should consider a more thorough review, like Math Workout for the GMAT, from The Princeton Review. This book will enable you to see where you need the most work. Always keep in mind, though, that if your purpose is to raise your GMAT score, it’s a waste of time to learn math that won’t be tested.


儘管我們可以向您展示對於GMAT最重要的數學原理,但本書無法代替數學的基礎知識。 我們發現,即使大多數人不記得很多高中數學,他們也會迅速重新學習。 如果您生鏽了,我們的練習和示例將刷新您的記憶,但是如果您對以下各章有嚴重的困難,則應考慮進行更徹底的複習,例如《普林斯頓評論》的GMAT的數學練習。 這本書將使您了解最需要工作的地方。 不過,請始終牢記,如果您的目的是提高GMAT分數,那將是浪費時間來學習未經測試的數學。


Try the following problem:

How many even integers are between 17 and 27 ?






This is an easy GMAT question. Even so, if you don’t know what an integer is, the question is impossible to answer. Before moving on to arithmetic, you should make sure you’re familiar with some basic terms and concepts. This material isn’t difficult, but you must know it cold. (The answer, by the way, is (C).)









這是一個簡單的GMAT問題。 即使這樣,如果您不知道整數是什麼,也無法回答這個問題。 在繼續進行算術運算之前,您應該確保自己熟悉一些基本術語和概念。 這種材料並不難,但您必須知道它很冷。 (順便說一句,答案是(C)。)


Integers are the numbers we think of when we think of numbers. Integers are sometimes called whole or natural numbers. They can be negative or positive. They do not include fractions. The positive integers are: 

1, 2, 3, 4, 5, etc.


The negative integers are:


–1, –2, –3, –4, –5, etc.


Zero (0) is also an integer. It is the only number that is neither positive nor negative. Sometimes the GMAT will reference a number that is not positive, or non-negative. That is another way to refer to zero as a potential part of a question.


Positive integers become greater as they move away from 0; negative integers become lesser. Look at this number line:


整數是我們想到數字時想到的數字。 整數有時稱為整數或自然數。 它們可以是負面的也可以是正面的。 它們不包括分數。 正整數是:







零(0)也是一個整數。 這是唯一既不是正數也不是負數的數字。 有時,GMAT會引用非正數或非負數。 這是將零稱為問題潛在部分的另一種方法。


正整數隨著遠離0而變大; 負整數變小。 看一下這個數字行:

2 is greater than 1, but –2 is less than –1.


Positive and Negative

Positive numbers lie to the right of zero on the number line. Negative numbers lie to the left of zero on the number line.


There are three rules regarding the multiplication of positive and negative numbers:


正數位於數字線上零的右邊。 負數位於數字線上零的左邊。



positive × positive = positive

positive × negative = negative

negative × negative = positive




If you add a positive number and a negative number, subtract the number with the negative sign in front of it from the positive number.


4 + (–3) = 1


If you add two negative numbers, you add them as if they were positive, and then put a negative sign in front of the sum.


–3 + –5 = –8




4 +(–3)= 1




–3 + –5 = –8


There are ten digits:


0, 1, 2, 3, 4, 5, 6, 7, 8, 9


All integers are made up of digits. In the integer 246, there are three digits: 2, 4, and 6. Each of the digits has a different name:


6 is called the units (or ones) digit.


4 is called the tens digit.


2 is called the hundreds digit.


A number with decimal places is also composed of digits, although it is not an integer. In the decimal 27.63, there are four digits:


2 is the tens digit.


7 is the units digit.


6 is the tenths digit.


3 is the hundredths digit.






所有整數均由數字組成。 在整數246中,有3個數字:2、4和6。每個數字都有不同的名稱:








儘管不是整數,但帶小數位的數字也由數字組成。 在十進制27.63中,有四個數字:










If an integer cannot be divided evenly by another integer, the integer that is left over at the end of division is called the remainder. Thus, remainders must be integers.


如果一個整數不能除以另一個整數,則除法結束時剩下的整數稱為餘數。 因此,餘數必須是整數。

Odd or Even

Even numbers are integers that can be divided evenly by 2, leaving no remainder. Here are some examples:


–6, –4, –2, 0, 2, 4, 6, etc.


偶數是可以除以2的整數,不留餘數。 這裡有些例子:



Any integer, no matter how large, is even if its last digit is divisible by 2. Thus 777,772 is even.


Odd numbers are integers that cannot be divided evenly by 2. Put another way, odd integers have a remainder of 1 when they are divided by 2. Here are some examples:


–5, –3, –1, 1, 3, 5, etc.


Any integer, no matter how large, is odd if its last digit is not divisible by 2. Thus 222,227 is odd.


There are several rules that always hold true for even and odd numbers:










even × even = even

odd × odd = odd 

even × odd = even

even + even = even

odd + odd = even

even + odd = odd







It isn’t necessary to memorize these, but you must know that the relationships always hold true. The individual rules can be derived in a second. If you need to know even × even, just try 2 × 2. The answer in this case is even, as even × even always will be.

不必記住這些,但您必須知道這些關係始終成立。 各個規則可以在一秒鐘內得出。 如果您需要知道偶數×偶數,請嘗試2×2。在這種情況下,答案是偶數,因為偶數×偶數總是如此。

Consecutive Integers

Consecutive integers are integers listed in order of increasing value without any integers missing in between. For example, –3, –2, –1, 0, 1, 2, 3 are consecutive integers. Only integers can be consecutive.


Some consecutive even integers: –2, 0, 2, 4, 6, 8, etc.


Some consecutive odd integers: –3, –1, 1, 3, 5, etc.


Distinct Numbers

If two numbers are distinct, they cannot be equal. For example, if x and y are distinct, then they must have different values.


連續整數是按值遞增順序列出的整數,中間沒有任何整數丟失。 例如,–3,–2,–1、0、1、2、3是連續的整數。 只有整數可以是連續的。







如果兩個數字不同,則它們不能相等。 例如,如果x和y不同,則它們必須具有不同的值。

Prime Numbers

A prime number is a positive integer that is divisible only by two numbers: itself and 1. Thus 2, 3, 5, 7, 11, and 13 are all prime numbers. The number 2 is both the least and the only even prime number. Neither 0 nor 1 is a prime number. All prime numbers are positive.


Divisibility Rules

If there is no remainder when integer x is divided by integer y, then x is said to be divisible by y. Put another way, divisible means you can evenly divide the greater number by the lesser number with no remainder. For example, 10 is divisible by 5.


質數是一個只能被兩個數字整除的正整數:本身和1。因此2、3、5、7、11和13都是質數。 數字2既是最小也是唯一的偶數。 0和1都不是質數。 所有素數均為正。



如果整數x除以整數y時沒有餘數,則稱x可被y整除。 換句話說,可整除意味著您可以將較大的數除以較小的數而沒有餘數。 例如,10可被5整除。

Some Useful Divisibility Shortcuts:


An integer is divisible by 2 if its units digit is divisible by 2. Thus, 772 is divisible by 2.


An integer is divisible by 3 if the sum of its digits is divisible by 3. We can instantly tell that 216 is divisible by 3, because the sum of the digits (2 + 1 + 6) is divisible by 3.


An integer is divisible by 4 if the number formed by its last two digits is divisible by 4. The number 3,028 is divisible by 4, because 28 is divisible by 4.


An integer is divisible by 5 if its final digit is either 0 or 5. Thus, 60, 85, and 15 are all divisible by 5.


An integer is divisible by 6 if it is divisible by both 2 and 3, the factors of 6. Thus, 318 is divisible by 6 because it is even, and the sum of 3 + 1 + 8 is divisible by 3.


Division by zero is undefined. The test-writers won’t ever put a zero in the denominator. If you’re working out a problem and you find yourself with a zero in the denominator of a fraction, you’ve done something wrong. By the way, a 0 in the numerator is fine. Any fraction with a 0 on the top is 0.






如果整數的整數可以被3整除,則整數可以被3整除。我們可以立即得知216被3整除,因為數字的總和(2 +1 + 6)被3整除。






如果整數可以被2和3(因子6)整除,那麼它可以被6整除。因此,318被偶數整除是6的整數,並且3 +1 + 8的和可以被3整除。






Factors and Multiples

An integer, x, is a factor of another integer, y, if y is divisible by x. So, in other words, y = nx, where y, n, and x are all integers. For example, 3 is a factor of 15 because 15 = (3)(5). All the factors of 15 are 1, 3, 5, and 15.


The multiples of an integer, y, are all numbers 0, ±1y, ±2y, ±3y…, etc. For example, 15 is a multiple of 3 (3 × 5); 12 is also a multiple of 3 (3 × 4). When you think about it, most numbers have only a few factors, but an infinite number of multiples. The memory device you may have learned in school is “factors are few; multiples are many.”


Every integer greater than 1 is both its own greatest factor and least positive multiple.


如果y被x整除,則整數x是另一個整數y的因數。 因此,換句話說,y = nx,其中y,n和x都是整數。 例如,3是15的因數,因為15 =(3)(5)。 15的所有因子均為1、3、5和15。


整數y的倍數都是0,±1y,±2y,±3y…等。例如15是3的倍數(3×5); 12也是3的倍數(3×4)。 考慮一下,大多數數字只有幾個因素,但無窮多個倍數。 您可能在學校學習的存儲設備是“因素很少; 倍數很多。”



Least Common Multiples

If an integer, x, is divisible by two integers, n and m, then x is a common multiple of n and m. For example, 30 is a common multiple of 5 and 6. The lowest multiple that two numbers have in common is called the least common multiple. For our example, 30 is also the least common multiple of 5 and 6.

The most straightforward way to find a least common multiple is to simply start listing the positive multiples of both integers. When you find a number that is on both lists, that number is the least common multiple.


For example, here’s how you find the least common multiple of 4 and 6.


Multiples of 4: 4, 8, 12, 16, 20,…


Multiples of 6: 6, 12,…


Since 12 is on both lists, 12 is the least common multiple of 4 and 6.


如果整數x可被兩個整數n和m整除,則x是n和m的公倍數。 例如,30是5和6的公倍數。兩個數字共有的最低倍數稱為最小公倍數。 在我們的示例中,30也是5和6的最小公倍數。

查找最小公倍數的最直接方法是簡單地開始列出兩個整數的正倍數。 當您在兩個列表中都找到一個數字時,該數字是最小公倍數。




4的倍數:4、8、8、12、16、20 …





Greatest Common Factor

If two integers, n and m, are both divisible by an integer, x, then x is a common factor of n and m. For example, 6 is a common factor of both 12 and 18. The highest factor that two numbers have in common is referred to as the greatest common factor. For our example, 6 is also the greatest common factor of 12 and 18.


The most straightforward way to find a greatest common factor is to simply list the factors of both numbers. Then you just need to find the greatest number that is on both lists.


For example, here’s how you find the greatest common factor of 12 and 18.


Factors of 12: 1, 2, 3, 4, 6, 12


Factors of 18: 1, 2, 3, 6, 9, 18


Since 6 is the greatest number on both lists, 6 is the greatest common factor of 12 and 18.


如果兩個整數n和m都可被整數x整除,則x是n和m的公因數。 例如,6是12和18的公因子。兩個數字共同具有的最高因子稱為最大公因子。 在我們的示例中,6也是12和18的最大公因子。


查找最大公因數的最直接方法是簡單地列出兩個數字的因數。 然後,您只需要查找兩個列表中的最大數字即可。









Prime Factors

If an integer, x, that is a factor of an integer, y, is also prime, then x is called a prime factor of y. For example, 3 and 5 are prime factors of 15.


To find the prime factors of an integer, use a factor tree:


如果作為整數y的因數的整數x也是素數,則x稱為y的質數因數。 例如,3和5是15的素因子。



A: 23. Remember, neither 0 nor 1 is prime.

All positive integers greater than 1 have unique prime factorizations, a fact that the GMAT frequently tests. So, it doesn’t matter which pair of factors you start with when you use the factor tree.

所有大於1的正整數都有唯一的素因式分解,這是GMAT經常測試的事實。 因此,使用因子樹時從哪對因子開始都沒關係。

The prime factorization of 12 is 2 × 2 × 3.


Absolute Value

The absolute value of a number is the distance between that number and 0 on the number line. The absolute value of 6 is expressed as .


數字的絕對值是該數字與數字線上的0之間的距離。 6的絕對值表示為。

 |6|= 6

 |-5|= 5

Challenge Question #1

x and y are integers less than 60 such that x is equal to the sum of the squares of two distinct prime numbers, and y is a multiple of 17. Which of the following could be the value of x – y ?









During the test, you’ll be able to see test instructions by clicking on the “Help” button at the bottom of the screen. However, to avoid wasting time reading these during the test, read our version of the instructions for Problem Solving questions now:


Problem Solving Directions: Solve each problem and choose the best of the answer choices provided.


Numbers: This test uses only real numbers; no imaginary numbers are used or implied.


Diagrams: All problem solving diagrams are drawn as accurately as possible UNLESS it is specifically noted that a diagram is “not drawn to scale.” All diagrams are in a plane unless stated otherwise.


在測試過程中,您可以通過點擊屏幕底部的“幫助”按鈕來查看測試說明。 但是,為了避免在測試期間浪費時間閱讀這些內容,請立即閱讀我們的“解決問題”說明版本:




數字:此測試僅使用實數; 沒有使用或暗示虛數。


圖表:所有解決問題的圖表均應盡可能準確地繪製,除非特別指出,圖表“未按比例繪製”。 除非另有說明,否則所有圖都在平面上。


Without a review of the basic terms and rules of math tested on the GMAT, you won’t be able to begin to do the problems.





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