Question

It is said that once a Chinese emperor decided to divide a number of horses among his generals such that

(a) if they are divided equally among 3 of his generals, 2 horses will remain;
(b) if they are divided equally among 5 of his generals, 1 horse will remain;
(c) and if they are divided equally among 7 of his generals, 6 horses will remain.


I. Find the smallest number of horses that satisfies this condition.



II. Write an explicit rule for each of part (a), (b) and (c).



III. Write an explicit rule for the number of horses that can be divided in such a way.



IV. Write a recursive rule for the explicit rule found in part III.

Answer 1

Using the three conditions, the smallest number of horses that can be shared among the generals is 41

The smallest number of horses

The conditions are given as:

• A remainder of 2, if divided among 3 generals
• A remainder of 1, if divided among 5 generals
• A remainder of 6, if divided among 7 generals

The numbers that can divide 5 and leave a remainder of 1 are:

11, 16, 21, 26, 31, 36, 41, 56, 61, 66, 71......

Of all these numbers, the smallest number that satisfy the three conditions is: 41

i.e.

41/3 = 13 R 2

41/5 = 8 R 1

41/7 = 5 R 6

Hence, the smallest number of horses is 41

The explicit rule for each condition

Let the number of horses a general gets be x.

The explicit rule for each condition can be represented as:

• Condition (a): Total = 3x + 2
• Condition (b): Total = 5x + 1
• Condition (c): Total = 7x + 6

The explicit and the recursive rule

Using the condition (b), we have:

Total = 5x + 1

Rewrite as a function

f(x) = 5x + 1

When x = 0, we have:

f(0) = 1

For other values of x, we have:

f(1) = 6

f(2) = 11

f(3) = 16

Rewrite as:

f(0) = 1

f(1) = 5 + 1 = 5 +f(0)

f(2) = 5 + 6 = 5 + f(1)

f(3) = 5 + 11 = 5 + f(2)

Express 2 as 3 - 1

f(3) = 5 + f(3 - 1)

Express 3 as x

f(x) = 5 + f(x - 1)

Hence, the recursive function is f(x) = 5 + f(x - 1) where f(0) = 1

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