We could see that 15 is the multiple of 3,5, and the remainder will be 1 if it is divided by 7
70 is the multiple of 7,5, and the remainder will be 1 if it is divided by 3
21 is the multiple of 3,7, and the remainder will be 1 if it is divided by 5
So the remainder will be 2 if 21*2 is divided by 5, and the remainder will be 4 if 15*4 is divided by 7
21*2+15*4+70=172
172/105=1……67
So x=105a+67=13b+6
Because a,b are non-negetive integers
(105a+61)is a multiple of 13
So 4 is the smallest number of a
So x=481+1365k(k is a non-negetive integer)
(3s1 Cao Wei)

(II) 射雕英雄傳裡寫道，有一次黃蓉因瑛姑而感到氣急敗壞，正欲反唇相譏之時，

念頭一轉，便用竹杖在㆞㆘細沙㆖寫了三道算題，裡面便有一道題目與餘數問題相關。這題目

是這麼出的：今有物，不知其数，三三数之，剩二，五五数之，剩三，七七数之，剩二，问物几何？」

ANS:

Let the number be x

Because the remainder is 2 when x is divided by 3 and 7,the remainder is 2 when x is divided by 21

x=21a+2=5b+3(a,b are non-negetive integers)
21a-1=5b
So the smallest value of a is 1
So x=23+105k(k is a non-negetive integer)
(3s1 Cao Wei)

The method 1) Give a list of the multiple of 3 with a remainder of 2 2,5,8,11,14,17…… 2) Give a list of the multiple of 5 with a remainder of 3 8,13,18,23…… 3) So 8 will be the fist number and f(x)=8+15x(x>=0) 4) The list of f(x) is 8,23,38,43,…… 5) Give a list of the multiple of 7 with a remainder of 2 2,9,16,23,30....... 6) So the second number is 23 and g(x)=23+105x(x>=0) because the total number is between 1000 and 1100, when x = 10, g(x)=1073

(Lu Zhen 3S1 14)

Please enter the number of answers to be displayed and click UPDATE:

April Online Questions:## The following are two Remainder Theorem related questions that appear in the Chinese Literature/Novel.

## Please contribute ONE POSSIBLE method to find their solution(s).

## Remainder to state your class and name at the end of it.

## (I) 韓信點兵又稱為中國剩餘定理，相傳漢高祖劉邦問大將軍韓信統御兵士多少，韓信答說，每3人一列餘1人、5人一列餘2人、7人一列餘4人、13人一列餘6人……。劉邦茫然而不知其數。

## ANS:

## Let the number of the soldiers be x

We could see that 15 is the multiple of 3,5, and the remainder will be 1 if it is divided by 770 is the multiple of 7,5, and the remainder will be 1 if it is divided by 3

21 is the multiple of 3,7, and the remainder will be 1 if it is divided by 5

So the remainder will be 2 if 21*2 is divided by 5, and the remainder will be 4 if 15*4 is divided by 7

21*2+15*4+70=172

172/105=1……67

So x=105a+67=13b+6

Because a,b are non-negetive integers

(105a+61)is a multiple of 13

So 4 is the smallest number of a

So x=481+1365k(k is a non-negetive integer)

(3s1 Cao Wei)

## (II) 射雕英雄傳裡寫道，有一次黃蓉因瑛姑而感到氣急敗壞，正欲反唇相譏之時，

## 念頭一轉，便用竹杖在㆞㆘細沙㆖寫了三道算題，裡面便有一道題目與餘數問題相關。這題目

## 是這麼出的：今有物，不知其数，三三数之，剩二，五五数之，剩三，七七数之，剩二，问物几何？」

## ANS:

## Let the number be x

## Because the remainder is 2 when x is divided by 3 and 7,the remainder is 2 when x is divided by 21

x=21a+2=5b+3(a,b are non-negetive integers)21a-1=5b

So the smallest value of a is 1

So x=23+105k(k is a non-negetive integer)

(3s1 Cao Wei)

## 韩信点兵

淮安民间传说着一则故事——“韩信点兵”，其次有成语“韩信点兵，多多益善”。韩信带1500名兵士打仗，战死四五百人，站3人一排，多出2人；站5人一排，多出3人；站7人一排，多出2人。韩信马上说出人数：1073。The method

1) Give a list of the multiple of 3 with a remainder of 2

2,5,8,11,14,17……

2) Give a list of the multiple of 5 with a remainder of 3

8,13,18,23……

3) So 8 will be the fist number and f(x)=8+15x(x>=0)

4) The list of f(x) is

8,23,38,43,……

5) Give a list of the multiple of 7 with a remainder of 2

2,9,16,23,30.......

6) So the second number is 23 and g(x)=23+105x(x>=0)

because the total number is between 1000 and 1100, when x = 10, g(x)=1073

(Lu Zhen 3S1 14)

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