## 1 重新认识余数

If  x and y are positive integers, there exist unique integers q and r, called the quotient and remainder, respectively   such that y=xq+r and 0≤ r＜x .

For example, 28 is divided by 8, the quotient is 3 and the remainder is 4 since 28=8×3+4.

A.     在y=qx+r中  r的范围是在[0, x）区间， 区间作为考点。

B.      在上式中 ，随着y每增加1 那么对应的r 要+1  直到到一个循环周期x （由 0 1 2 3…x-1作为一个完整的循环） r是有周期性  周期性作为考点。

C.      。本身y=qx+r 作为性质考点。在给出的例子中28=8×3+4 种作为函数考点

D.     同余问题

## 实战演习

### A考点。 区间考点

14 mod 6 =2 means14 is divided by 6 the remainder is 2 ,  x mod 6 is ?

1 x mod 3=x mod12

2 x mod 4=2

【解释】mod 就是取余函数 也就是问 x÷6的余数是多少？

1à x被3除的余数=x被12除的余数  说明余数＜3  （当余数＞3的时候 比如被12除余数是4 那么被3除余数=1 ≠4 ）  余数=0 1 or 2  not sufficient

2àx=4m+2   4m+2除6 未知m  not sufficient

1+2à x mod 12=x mod 3=x mod4   (因为 3 and 4都是12的factor )

so  x mod 12=2 =x mod 6  Sufficient

【答案】C

### B考点：周期性考点

What is the remainder when 2204 is divided by 10?

【解释】

2的n次方的个位数 是2 4 8 6  四个一循环

204 ÷4 余数=0   （用指数÷循环周期 看余数）

【答案】6

If x is a positive integer, is the remainder 0 when  is divided by 10?
(1)x = 4n + 2, where n is a positive integer.
(2)x
4

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 statement TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D EACH statement ALONE is sufficient.

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

【解释】

1à3对应的是3 9 7 1   4n+2被4除余数是2 对应9   9+1 除以10 余数是0  sufficient

2à未知3x的个位数多少 +1后依然不确定 not sufficient

【答案】A

### C 考点： 性质 函数考点

If x is a positive integer, when x divided by 3 the remainder is 2 what is the remainder when x²+3x+1 divided by 3?

A.     1

B.     2

C.     3

D.     4

E.     0

【解释】

x=3n+2   x²+3x+1=（3n+2）²+3（3n+2）+1=9n²+12n+4+9n+6+1=9n²+21n+11

divided by 9   9n²  21n 都能整除3    11÷3 余数=2

【答案】B 2

What is the remainder when the positive integer n is divided by 3?
(1)The remainder when n is divided by 2 is 1.
(2)The remainder when n + 1 is divided by 3 is 2.

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 statement TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D EACH statement ALONE is sufficient.

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

【解释】求n除3的余数

1àn被2除余1 写成 n=2x+1的形式  2x+1除以3   余数由x决定  not sufficient

2àn+1=3y+2(这里注意写成y 和x不同的字母以免混淆)   n=3y+1   3y+1÷3 余1 sufficient

【答案】B

### D 同余问题

When positive integer x is divided by 5, the remainder is 3; and when x is divided by 7, the remainder is 4. When positive integer y is divided by 5, the remainder is 3; and when y is divided by 7, the remainder is 4. If xy, which of the following must be a factor of x - y?

A.     12

B.     15

C.     20

D.     28

E.     35

【解释】

x=5m+3=7n+4     第一步 按照题意写出等式关系 （2个未知数）

14-15=-1   ∴ n=2  m=3 第二步 带入一个可以看出的数值 让等式成立（试数法）求出未知数的可行解

5and 7 最小公倍数=35    第三步 求出两个等式关系的除数的最小公倍数

x=35q+18

【答案】E