【小站教育】GMAT数学笔记

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GMAT 数学笔记

GMAT 数学备考关键词 一、

知识点：准确掌握 二、词汇、表达法：读懂题目 三、熟练：平均两分钟一道题

考试相关问题 一、时间与题量 二、题型 三、机经与换题库 四、其它

If a and b are positive integers such that a – b and a/b are both even integers, which of the following must be an odd integer? （A） a/2 （B） b/2 （C）(a+b)/2 （D） (a+2)/2 （E） (b+2)/2

If M is the least common multiple of 90, 196, and 300, which of the following is NOT a factor of M? (A) 600

(B)700 (C) 900 (D) 2,100 (E) 4,900

复习注意事项 *战略上重视 *初等数学的思维 *解法力求稳妥清晰 *把握好 DS 题型 *熟练重于技巧 推荐复习步骤

*知识点查缺补漏 *背熟词汇 *复习课上所学

*OG，及其它相关资料

*机经：www.chasedream.com

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第一章 算术

1. integer (whole number): 整数

* positive integer：正整数，从 1 开始，不包括 0。 2. odd & even number 奇数与偶数

* 凡整数均具有奇偶性，如－1 是奇数，0 是偶数。 * 奇+奇=偶，奇+偶=奇…

若干个整数相乘，除非都是奇数，其乘积才会是奇数… 例： If a and b are positive integers such that a – b and which

of the following must be an odd integer?

a ? b a ? 2 b ? 2 a b (A） (B） (C)(D)(E)

2 2 2 2 2

3. prime number & composite number 质数与合数

a

are both even integers, b

* A prime number is a positive integer that has exactly two different positive divisors,1 and itself.

* A composite number is a positive integer greater than 1 that has more than two divisors.

* The numbers 1 is neither prime nor composite, 2 is the only even prime number.

4. factor(divisor) & prime factor 因子和质因子

* 一个数能被哪些数整除，这些数就叫它的因子（因数、约数）。 * 因子里的质数叫质因子（数）。

例 1： If n=4p, where p is a prime number greater than 2, how many different

positive even divisors does n have, including n?

(A) 2 (B) 3 (C) 4 (D) 6 (E) 8 例 2：If the integer n has exactly three positive divisors, including 1 and n, how

many positive divisors does n2 have?

(A) 4 (B) 5 (C) 6 (D) 8 (E) 9 例 3：1225 有几个因子？

例：What is the greatest prime factor of 2100 - 296? (A) 2

(B) 3

(C) 5

(D) 7

(E) 11

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例：A positive integer n is said to be “prime-saturated” if the product of all the different positive prime factors of n is less than the square root of n. What is the greatest two-digit prime-saturated integer?

(A) 99 (B) 98 (C) 97 (D) 96 (E) 95

5. the greatest common divisor (GCD)& the least multiple(LCM) 最大公约数和最小公倍数

common

例：If M is the least common multiple of 90, 196, and 300, which of the

following is NOT a factor of M?

(A) 600 (B)700 (C) 900 (D) 2,100 (E) 4,900

例：What is the lowest positive integer that is divisible by each of the integers 1 through 7,inclusive?

(A) 420 (B) 840 (C) 1,260 (D) 2,520 (E) 5,040 6. decimals & fractions 小数和分数

*相关词汇：reaccuring decimal ; terminating decimal ; numerator ;

denominator ; improper fracion ; mixed number *整数位与分位： 后面加 s 的是整数位（小数点前面的某位），加 th 或 ths 的是 分位（小数点后面的某位），如 tens 是十位数，而 tenth 是十分位 *What is the fractional part of ….这样的表达法意为“谁的几分之几” *小数和分数的互相转换： 例 1： 0．373737…=? (将其转换成一个分数)

例 2：Which of the following fractions has a decimal equivalent that is a terminating decimal?

(A) 10/189 (B) 15/196 (C) 16/225 (D) 25/144 (E) 39/128 7. consecutive numbers 连续数 例 1：In an increasing sequence of 10 consecutive integers, the sum of the first 5

integers is 560. What is the sum of the last 5 integers in the sequence? (A) 585 (B) 580 (C )575 (D)570 (E) 565

例 2：If n is an integer greater than 6, which of the following must be divisible by 3?

(A) n(n+1)(n-4) (D) n(n+4)(n-2)

(B) n(n+2)(n-1) (E) n(n+5)(n-6)

(C) n(n+3)(n-5)

8. divisibility & remainder 整除及余数问题

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* 一个数是否能够被 5 整除，只要看它的最后一位（是 0 或 5）。 * 一个数是否能够被 4 整除，只要看它的后两位（是否是 4 的倍数）。 * 一个数是否能够被 8 整除，只要看它的后三位（是否是 8 的倍数）。 * 一个数能否被 3 整除，取决于各位之和能否被 3 整除。 * 一个数能否被 9 整除，也取决于各位之和能否被 9 整除。 * 0 能被所有数整除。

* 余数包括 0，如 24 除以 6，商为 4 余数为 0。 例：1912 257 的个位数字是几？

例：If s and t are positive integers such that s ? 64.12 ,which of the following could be the

t

remainder when s is divided by t? (A) 2

(B) 4

(C) 8

(D) 20

(E) 45

9. 数字问题

例：1001 位数字组成的数，任意相邻的两位数字组成的数能被 17 或 23 整除， 这个 1001 位的数字以 6 开头，则它的最后六位是（ ） 10. 算术部分的几种常用方法

*参数法 例：两个两位数个位与十位恰好颠倒，问下面哪个不能是两数之和？ A.181，B.121，C.77，D.132，E.154

解法：设两数分别为 ab 和 ba，则(ab)+(ba)=(10a+b)+(10b+a)=11(a+b)，即和 必为 11 的

倍数，答案为 A。

*代数法 *试错法 例：

□△

× △□ The product of the two-digit numbers above is the three-digit number □◇□, where □,△,and◇ □are △＜10, what is the two-digit number △? three different nonzero digits. If □×(A) 11 (B) 12 (C) 13 (D) 21 (E) 31

第二章 代数

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1． Quadratic equations: 一元二次方程

ax2+bx+c=0

? b ??b2 ? 4ac x 1,2 ? 2a

但一般更常用的是因式分解法：

x2-2x-3=0 (x-3)(x+1)=0 x1=3, x2=-1

2. Simultaneous linear equations: 多元一次方程组 * 基本方法：消元法。 例 1：3x+y=5 (1) 2x+y=4 (2)

(1)－(2), 消去 y, 得 x=1,y=2

* 注意：并不是任何二元一次方程组都有唯一解。

例 2: 3x+y=5 (1)

6x+2y=10 (2) 上述方程有无穷多组解。 因此，方程的数量须等于未知数的数量，此时多元一次方程有唯一的一组解。 3. Simultaneous quadratic equations: 二元二次方程组 一般只考如下形式：

a1x+b1y=c1 (1)

a2x2+b2x+a3y2+b3y=c2 (2)

即其中一个方程为一次。这种形式等价于一元二次方程，把（1）代入（2）即可。 4. Inequalities: 不等式

*不等式部分不会像中国高考那样考推导、证明，注意两边乘以负数变号等最基 本原则即可。

5. Arithmetic sequence: 等差数列 an=a1+(n-1)d sn=(a1+an)n/2 n=(an-a1)/d +1

6．Geometric sequence: 等比数列

an=a1qn-1

1 ? q n s n ? a1 ? 1 ? q

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a1

当∣q∣<1 时， s ?? ? 1 ? q

1 11 1例： ? 2 ? 3 ? ? ? ? ? ？

2 2

2 2

例： 0．373737…=? (将其转换成一个分数) 7．Sets: 集合 例 1：全班 50 个人，选音乐课的有 20 人，选体育课的有 18 人，两课都选的有 5 人，问两课都没选的几人？ 例 2： A marketing firm determined that, of 200 households surveyed, 80 used neither Brand A nor Brand B soap, 60 used only Brand A soap, and for every household that used both brands of soap, 3 used only Brand B sop. How many of the 200 households surveyed used both brands of soap? (A) 15 (B)20 (C)30 (D)40 (D)45 例 3：五个人排队，甲不能在首位，乙不能在末位，有几种不同的排法？

第三章 几何

1． Lines & planes 直线与平面

* 两直线平行并为第三条直线所截后，相应角的关系。 * 直线与平面的关系。

例：If n distinct planes intersect in a line, and another line L intersects one of these planes in a single point, what is the least number of these n planes that L could intersect?

(A) n (B) n－1 (C) n－2 (D) n/2 (E)(n－1)/2 2. Triangles 三角形 * 勾股定理：a2+b2=c2

* 构成三角形的条件：两边之和大于第三边。 * 三角形内部边和角的关系：大边对大角。 3. Quadrilaterals 四边形

* parallelogram(平行四边形) : 面积=a×h; 周长=2（a+b） * rectangle(矩形) : 面积=a×h; 周长=2（a+b） * square(正方形) : 面积=a2 ; 周长=4a * trapezoid(梯形) : 面积=（a+b）×h/2

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4. Circles 圆 * 面积=πR2 * 周长=2πR

5. Polygons 多边形 * 多边形内角和：（n-2）180o

6. Rectangular Solids 长方体 * 体积=a×b×c * 表面积=2（a×b+b×c+c×a） 7. Cubes 正方体 * 体积=a3

* 表面积=6a2

8. Cylinders 圆柱 * 体积=πR2h

* 表面积=2πR2+2πR×h 例：一个圆锥内接于一个半球，圆锥的底面与半球的底面重合，则圆锥的高与半 球的半径的比是多少？

9. Coordinate Geometry 解析几何

* 直线的标准方程：y=kx+b ；即斜截式，其中 k 为斜率 slope，b 为 y 轴截距

y-intercept

* 斜率的计算：K= (Y2-Y1)/( X2-X1) * 两点或一点加斜率确定一条直线。 * 两直线垂直，其斜率的乘积为－1。

第四章 统计

1. arithmetic mean (average) 算术平均值 1 n E= ? ai

n i ?1

2. median 中位数

* The median is the middle value of a list when the numbers are in order. * 先排序，后取中。 3. mode 众数

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* The mode of a list of numbers is the nmuber that occurs most frequently in the list.

* A list of numbers may have more than one mode. 4. expectation 期望 * 期望就是算术平均值。 5. deviation 偏差

di=ai-E 6. variance 方差

1 n 2D ?? ? ?ai ? E ??n i ?1

7. standard deviation 标准差

? ?? D

例：Ⅰ.72,73,74,75,76

Ⅱ.74,74,74,74,74 Ⅲ.62,74,74,74,89

The data sets Ⅰ, Ⅱ,and Ⅲ above are ordered from greatest standard deviation to least standard deviation in which of the following ? (A) Ⅰ,Ⅱ, Ⅲ (B) Ⅰ, Ⅲ,Ⅱ (C) Ⅱ, Ⅲ, Ⅰ (D) Ⅲ, Ⅰ,Ⅱ Ⅱ, Ⅰ

8. range 范围

* 最大数减去最小数所得的差就是该组数据的范围。 例 1：150, 200, 250, n

(E) Ⅲ,

Which of the following could be the median of the 4 integers listed above?

Ⅰ. 175 Ⅱ. 215 Ⅲ. 235

(A) Ⅰonly (B) Ⅱonly (C) Ⅰand Ⅱonly (D) Ⅱand Ⅲ only (E) Ⅰ,Ⅱ,and Ⅲ 例 2：The least and greatest numbers in a list of 7 real numbers are 2 and 20,respectively. The median of the list is 6,and the number 3 occurs most often in the list. Which of the following could be the average of the numbers in the list?

Ⅰ. 7 Ⅱ. 8.5 Ⅲ. 10.5

(A)Ⅰonly (B) ⅠandⅡonly (C) Ⅰand Ⅲ only (D) Ⅱand Ⅲ only (E) Ⅰ,

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Ⅱ,and Ⅲ

第五章 数据充分性题

*每道 DS 题的选项都是固定的：

A Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C BOTH statements TOGETHER are sufficient, but NEITHER statement

ALONE is sufficient.

D Each statement ALONE is sufficient.

E Statement (1) and (2) TOGETHER are not sufficient.(additional data are

needed).

* DS 题的本质是一种判断型的选择题，并非判断正误，而是判断根据条件给的信 息能否回答主题干里提出的问题。 *要注意的几大问题： <1> 唯一性 例：x=?

（1）x=2 （2）x2=4 <2> 否定性 例：x>0? （1）x2>0 （2）x3<0

<3> 不矛盾性 例：A，B 两车在长直道路上相对行驶，现距离为 500 英里，问多长时间后相遇？ (1) 其中一辆速度为 200 英里每小时。 (2) 其中一辆速度为 300 英里每小时。

<4> 独立性 例：x>0? （1）√x=5

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（2）x3<0

<1> If n is an integer, is n+1 odd? (1) n+2 is an even integer. (2) n-1 is an odd integer.

<2> In △PQR,if PQ=x, QR=x+2, PR=y, which of the three angles of △PQR

has the greatest degree measure?

(1) y=x+3 (2) x=2

<3> Tom, Jane, and Sue each purchased a new house . The average (arithmetic mean )price of the three houses was $120,000.What was the median price of the three houses?

(1) The price of Tom’s house was $110,000. (2) The price of Jane’s house was $120,000. <4> 3.2□△6, □=?

(1) 3.2□△6 四舍五入到十分位后是 3.2。

(2) 3.2□△6 四舍五入到百分位后是 3.24。

<5>If °represents one of the operations +, -,and ×,is k°(l +m)=(k°l)+(k°m)for all numbers k, l , and m?

(1) k°1 is not equal to 1°k for some numbers k. (2) °represents subtraction.

<6>On Jane's credit card account, the average daily balance for a 30-day billing cycle is the average of the daily balances at the end of each of 30 days. At the beginning of a certain 30-day billing cycle, Jane's credit card account had a balance of $600. Jane made a payment of $300 on the account during the billing cycle. If no other amounts were added to or subtracted from the account during the billing cycle, what was the average daily balance on Jane’s account for the billing cycle?

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(D)Ⅲ only (E) Ⅰand Ⅲ

<3> What is the greatest prime factor of 2100 - 296? (A) 2

(B) 3

(C) 5

(D) 7

(E) 11

<4> A positive integer n is said to be “prime-saturated” if the product of all the different

positive prime factors of n is less than the square root of n. What is the greatest two-digit

prime-saturated integer?

(A) 99 (B) 98 (C) 97 (D) 96 (E) 95

<5> If 1050 – 74 is written as an integer in base 10 notation, what is the sum of the digits in

that integer?

（A）424 （B）433 （C）440 (D) 449 (E) 467

<6> If P and Q are different points in a plane, the set of all points in this plane that are closer to P than to Q is

(A) the region of the plane on one side of a line (A) the interior of a square

(B) a wedge-shaped region of the plane

(C) the region of the plane bounded by a parabola (D) the interior of a circle

<7> A researcher computed the mean, the median, and the standard deviation for a set of performance scores. If 5 were to be added to each score, which of these three statistics would change? (A)Ⅰonly Ⅱ,and Ⅲ

(B) Ⅱonly

(C) Ⅰand Ⅱonly

(D) Ⅱand Ⅲ only (E) Ⅰ,

<8> Test y physics biology history chemistry math

onya's score 90 96 89 88 80 Mean score 84 86 81 85 88 tandard deviation 2 5 4 3 4 Th e cha rt abo

ve shows data for five tests that Sonya took. On which of the five tests did she

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score highest relative to the rest of the test takers?

(A) math (B) biology (C) physics (D) history (E) chemistry <9> Some people in New York express 2/8 as 8th Feb and others express 2/8 as 2nd Aug. This can be confusing as when we see 2/8, we don’t know whether it is 8th Feb or 2nd Aug. However, it is easy to understand 9/22 or 22/9 as 22nd Sept, because there are only 12 months in a year. How many dates in a year can cause this confusion?

（A）48 （B) 53 （C）120 （D）132 （E）144

<10> Beginning in January of last year, Carl made deposits of $120 into his account on the 15th of each month for several consecutive months and then made withdrawals of $50 from the account on the 15th of each of the remaining months of 1ast year. There were no other transactions in the account last year. If the closing balance of Carl's account for May of last year was $2600, what was the range of the monthly closing balances of Carl's account last year?

(1) Last year the closing balance of Carl's account for April was less than $2625. (2) Last year the closing balance of Carl's account for June was less than $2675. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D. EACH statement ALONE is sufficient.

E. Statements (1) and (2) TOGETHER are NOT sufficient.

<11> At the beginning of a five-day trading week, the price of a certain stock was $10 per share. During the week, four of the five closing prices of the stock exceeded $10. Did the average closing price of the stock during the week exceed its price at the beginning of the week?

(1) The stock's closing price on Tuesday was the same as its closing price on Thursday. (2) The sum of the stock's highest and lowest closing prices during the week was 20. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D. EACH statement ALONE is sufficient.

E. Statements (1) and (2) TOGETHER are NOT sufficient.

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<12> If n is a positive integer and r is the remainder when (n-1)(n+1) is divided by 24, what is the value of r? (1) 2 is not a factor of n. (2) 3 is not a factor of n.

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D. EACH statement ALONE is sufficient.

E. Statements (1) and (2) TOGETHER are NOT sufficient.

<13> 三人独立地去破译一个密码，他们能译出的概率分别为 1/5,1/3,1/4, 求此密 码被译出的概率。

<14> 某市共有 10000 辆自行车，其牌照号码从 00001 到 10000，求偶然遇到的 一辆自行车，其牌照号码中有数字 8 的概率。 <15> 用 0,2,4,6,9 这五个数字可以组成数字不重复的五位偶数共有多少个？ <16>有 5 个队伍参加了足球联赛，任意两队之间都要进行主客场各一场比赛，问总 共将有多少场比赛？

<17> 甲，乙，丙，丁，戊五人并排站成一排，如果乙必须站在甲的右边（甲乙可 以不相邻），那么不同的排法共有多少种？ <18> 4 本不同的书分给 2 人，每人 2 本，不同的分法有多少种？

<19> 两把 keys,放到有 5 个 keys 的 keychain 中，相邻的概率为多少(分直线和环 形)？

<20> 6 张同排连号的电影票，分给 3 名男生和 3 名女生，如欲男女相间而坐，则

不同的分法数为多少？ <21> 有 4 对人，从中取 3 个人，不能从任意一对中取 2 个，问有多少种取法？ <22> 从 6 双不同的手套中任取 4 只，求其中恰有一双配对的概率。 <23> 3 封不同的信，有 4 个信箱可供投递，共有多少种投信的方法？ <24> 3 个打字员为 4 家公司服务，现在每家公司各有 1 份文件需要录入，问每个打 字员都收到文件的概率？

<25> A 发生的概率是 0.6，B 发生的概率是 0.5，问 A,B 都不发生的最大概率？ <26> 袋中有 a 只白球，b 只红球，依次将球一只只摸出，不放回。求第 k 次摸出 的是个白球的概率（1≤k≤a+b）。 <27> 掷一枚均匀硬币 2n 次，求出现正面 k 次的概率。

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<28> 有 4 组人，每组一男一女，从每组各取一人，问取出两男两女的概率？

<29> A certain roller coaster has 3 cars, and a passenger is equally likely to ride in any 1 of the 3 cars each time that passenger rides the roller coaster. If a certain passenger is to ride the roller coaster 3 times, what is the probability that the passenger will ride in each of the 3 cars? （A）0 （B）1/9（C）2/9 （D）1/3（E）1

<30>A gardener is going to plant 2 red rosebushes and 2 white rosebushes. If the gardener is to select each of the bushes at random, one at a time, and plant them in a row, what is the probability that the 2 rosebushes in the middle of the row will be the red rosebushes?

（A）1/12 （B）1/6 （C）1/5 （D）1/3 （E）1/2

<31> If a committee of 3 people is to be selected from among 5 married couples so that the committee does not include two people who are married to each other, how many such committees are possible?

（A）20 （B）40 （C）50 （D）80 （E）120

<32> How many seven-digit numbers contain the digit '7' at least once? <33> In how many ways can seven students A, B, C, D, E, F and G line up in one row if students B and C are always next to each other?

<34> There are 5 cars to be displayed in 5 parking spaces with all the cars facing the same direction. Of the 5 cars, 3 are red, 1 is blue, and 1 is yellow. If the cars are identical except for color, how many different display arrangements of the 5 cars are possible?

(A)20 (B) 25 (C) 40 (D) 60 (E) 125

<35> How many different 6-letter sequences are there that consist of 1 A, 2 B’s, and 3 C’s ?

(A)6 (B) 60 (C) 120 (D) 360 (E) 720

<36> In how many distinguishable ways can the seven letters in the word MINIMUM be arranged, if all the letters are used each time? (A) 7 (B) 42 (C) 420 (D) 840 (E)5040

<37> A photographer will arrange 6 people of 6 different heights for photograph by placing them in two rows of three so that each person in the first row is standing in front of someone in the second row. The heights of the people within each row must increase from left to right, and each person in the second

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row must be taller than the person standing in front of him or her. How many such arrangements of the 6 people are possible?

(A)S

(B) 6 (C) 9 (D) 24 (E)36

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