数论
133
Quantity A：Number of primes between 50 and 60
Quantity B：Number of primes between 80 and 90
A. Quantity A is greater.
B. Quantity B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the information given.
- x and y are prime numbers and x+y=18
Quantity A：xy
Quantity B：70
A. Quantity A is greater.
B. Quantity B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the information given.
- x is a positive integer. k is the remainder when x3-x is divided by 3.
Quantity A：k
Quantity B：1
135. 答案：B
**解析：**3的倍数肯定没问题，能被3整除，剩下的数字有两证情况 x=3n+1 和x=3n+2，分别带入原式子,结果均可以被3整除，x3-x=x(x-1)(x+1) x为正整数，所以x3-x肯定是3的倍数。
- x and y are integers greater than 5. x is y percent of x2
Quantity A：x
Quantity B：10
136. 答案：C
**解析：**根据题意有y%=x/x2=1/x，所以xy=100，又因为x和y都是大于5的整数，所以所以只能是x=10，y=10。正确答案选C。
139. 答案：ABC
**解析：**原式化简得w=(x2+3)y+x，分类讨论即可。
若x为odd，则x2+3为even，此时无论y是odd还是even，(x2+3)y都是even，w为odd。
若x为even，则x2+3为odd，此时再讨论：
- 若y为even，(x2+3)y为even，w为even
- 若y为odd，(x2+3)y为odd，w为odd
至此全部情况讨论完毕，AB很容易选出来，此题容易漏选C。
- When the even integer n is divided by 7, the remainder is 3.
Quantity A：The remainder when n is divided by 14.
Quantity B：10
循环规律
What is the remainder when is divided by 5?
1，3，9，7循环
- If 2, 4, 6, 9 are the digits of two 2-digit integers, what is the least possible positive difference between the integers?
代数
化简运算
If N is an integer and 99＜N2＜200, then N could have at most how many values?
正负都可以
If one number is chosen at random from the first 1,000 positive integers, what is the probability that the number chosen is a multiple of both 2 and 8?
125
集合
容斥原理
统计
- The range of the heights of the female students in a certain class is 13.2 inches, and the range of the heights of the male students in the class is 15.4 inches.
Which of the following statements individually provide(s) sufficient additional information to determine the range of the heights of all the students in the class?
Indicate all such statements.
A. The tallest male student in the class is 5.8 inches taller than the tallest female student in the class.
B. The median height of the male students in the class is 1.1 inches greater than the median height of the female students in the class.
C. The average (arithmetic mean) height of the male students in the class is 4.6 inches greater than the average height of the female students in the class.
假设r是奇数，那么t则为偶数。奇数的平方仍为奇数，所以p=r2+t=奇+偶=奇，A=-1的奇次幂=-1=B；假设r是偶数，那么t则为奇数。偶数的平方仍为偶数，所以p=r2+t=偶+奇=奇，A=-1的奇次幂=-1=B。所以答案就是C。
9*10+9+9=108
几何
看起来平行不代表平行
组合
124
How many integers between 1 and 1021are such that the sum of their digits is 2?
**解析：**可以这么想，一共有22位数，选2时，从21个位选一个数，有21种；1+1的情况，从21个数中选2个空填进去C221=210种，加起来231中。
125
There are 10 people in a room. If each person shakes hands with exactly 3 other people, what is the total number of handshakes?
**解析：**先不考虑重复，每个人都与三个人握手一共有10*3=30，但是一个人与另一个人握手相当于自己与另一个人和另一个人与自己握手，因此30要除以2。
126
How many positive integers less than 10,000 are such that the product of their digits is 210?
**解析：**210 = 2x5x3x7 = 5x6x7x1 = 5x6x7
Those are the only sets of digits we can use to for the numbers (any other combination of factors will have two digit factors).
Numbers using 2,5,3,7 = 4!
Numbers using 5,6,7,1 = 4!
Numbers using 5,6,7 (3-digit numbers) = 3!
Answer = 24+24+6 = 54。
127
In a group of 45 children, 60 percent of the children are boys, and 60 percent of the children are left-handed.
Quantity A：Number of boys who are left-handed
Quantity B：8