Imagine you are an old Chinese scholar teaching Proof and Numbersto Chinese princes. When you group
your pupils in rows of �five, two are left over. If you group them in eights, only one is left over. So you
try it with rows of nine, but now you are left with seven pupils. What do you think would happen if
you grouped your pupils in rows of eleven? And can you guess how many pupils you have?
You could do it by brute force
conditon 1: remainder of 2 when divided by 5 ...
7 12 17 22 27 32 ... 82 87 92 97 102 .....
condition 2: remainder of 1 when divided by 8
9 17 25 33 41 .... 89 97 105 ...
condition 3: remainder of 7 when divided by 9
16 25 34 .... 88 97 106 ...
ahhh, looks like 97 is in all three cases
So there could be 97 students
If grouped in rows of 11 , the remainder would be 9
A more rigorous method is to use the
"Chinese Remainder Theorem"
It is a rather complicated method and you must know how to manipulate expressions such as
a mod(b)
e.g. 47 = 2mod9
If you want to know more about the Chinese Remainder Theorem , here is a good YouTube for it
http://www.youtube.com/watch?v=3PkxN_r9up8
To tackle this problem, we can use a method called modular arithmetic to find a solution. Let's consider each scenario one by one and use the concept of remainders.
1. When you group the pupils in rows of five, two are left over: This implies that the total number of pupils leaves a remainder of 2 when divided by 5. In other words, the number of pupils can be expressed as 5n + 2, where n is an integer.
2. When you group the pupils in rows of eight, only one is left over: This suggests that the total number of pupils leaves a remainder of 1 when divided by 8. So, the number of pupils can be written as 8m + 1, where m is an integer.
3. When you group the pupils in rows of nine, seven are left over: This means that the total number of pupils leaves a remainder of 7 when divided by 9. Therefore, the number of pupils can be represented as 9k + 7, where k is an integer.
Now, let's analyze the scenario when you group the pupils in rows of eleven.
If we assume that the total number of pupils can be expressed as 11x + y, where x is an integer and y is the remaining amount, we need to find what y is.
Since all the given scenarios involve remainders, we can conclude that the remaining values (2, 1, and 7) are equivalent modulo 11. In other words, 2 ≡ 1 ≡ 7 ≡ y (mod 11).
Now, to find the value of y, we can start with a common multiple of 2, 1, and 7. Let's check if 2, 1, and 7 have a common multiple: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ...
From the multiples, we can observe that the smallest common multiple of 2, 1, and 7 is 14.
Therefore, y = 14 would be the remainder when the number of pupils is divided by 11.
To determine the total number of pupils, we now need to find a value for x that satisfies the other conditions mentioned earlier.
The equation representing the total number of pupils is 11x + 14.
We can start plugging in different values for x until we find a value that satisfies all the conditions (leaving remainders of 2 when divided by 5, 1 when divided by 8, and 7 when divided by 9).
Let's start by checking x = 1:
For x = 1, the total number of pupils would be 11 * 1 + 14 = 25.
When divided by 5, 25 leaves a remainder of 0 (not 2 as required).
Let's try another value, x = 2:
For x = 2, the total number of pupils would be 11 * 2 + 14 = 36.
When divided by 5, 36 leaves a remainder of 1 (not 2 as required).
Continuing this process, we can find that x = 7 is the first value that satisfies all the conditions.
For x = 7, the total number of pupils would be 11 * 7 + 14 = 91.
To summarize, if you group your pupils in rows of eleven, you would have 91 pupils.