Math Problem Statement
- This is an ancient puzzle problem: "Yen Kung, 1372. We have unknown number of coins. If you make 77 strings. of them, you are 50 coins short; but if you make 78 strings, it is exact. How many coins are there?" [Hint: If N is the number of coins, then N = 77x + 27 = 78y * 0.1 (25 points)
Solution
This is a classic problem involving simultaneous linear congruences. Here's how to solve it step by step:
Problem Restatement
We need to find the number of coins , which satisfies:
- (Equation 1)
- (Equation 2)
Where and are integers.
Step 1: Combine the equations
From Equation 1 and Equation 2, we equate the expressions for :
Rearranging gives:
Step 2: Solve using modular arithmetic
Rewriting the equation:
Since , the equation simplifies to:
Thus, , where is an integer.
Step 3: Find
Substituting into Equation 2:
Step 4: Verify minimal
For , . Verifying this:
-
If : Subtract 27: Dividing:
-
If : Dividing:
Thus, the solution is:
Follow-up Questions
- Can you solve a similar problem with different numbers, such as 85 strings and 86 strings?
- What does modular arithmetic tell us about the divisors of ?
- How would this change if the hint involved a remainder other than 27?
- Can you write a generalized formula for such problems using congruences?
- How can this problem be visualized geometrically or with number lines?
Tip: Modular arithmetic is an excellent tool for solving problems involving divisors and congr
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Math Problem Analysis
Mathematical Concepts
Modular Arithmetic
Simultaneous Linear Congruences
Number Theory
Formulas
N = 77x + 27
N = 78y
78y ≡ 27 (mod 77)
Theorems
Chinese Remainder Theorem (implicitly related)
Suitable Grade Level
Grades 10-12
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