Math Problem Statement

  1. 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 NN, which satisfies:

  1. N=77x+27N = 77x + 27 (Equation 1)
  2. N=78yN = 78y (Equation 2)

Where xx and yy are integers.

Step 1: Combine the equations

From Equation 1 and Equation 2, we equate the expressions for NN: 77x+27=78y77x + 27 = 78y

Rearranging gives: 78y77x=2778y - 77x = 27

Step 2: Solve using modular arithmetic

Rewriting the equation: 78y27(mod77)78y \equiv 27 \pmod{77}

Since 78mod77=178 \mod 77 = 1, the equation simplifies to: y27(mod77)y \equiv 27 \pmod{77}

Thus, y=27+77ky = 27 + 77k, where kk is an integer.

Step 3: Find NN

Substituting y=27+77ky = 27 + 77k into Equation 2: N=78y=78(27+77k)N = 78y = 78(27 + 77k) N=7827+7877kN = 78 \cdot 27 + 78 \cdot 77k N=2106+6006kN = 2106 + 6006k

Step 4: Verify minimal NN

For k=0k = 0, N=2106N = 2106. Verifying this:

  1. If N=77x+27N = 77x + 27: 2106=77x+272106 = 77x + 27 Subtract 27: 2079=77x2079 = 77x Dividing: x=27(integer, valid)x = 27 \quad \text{(integer, valid)}

  2. If N=78yN = 78y: 2106=78y2106 = 78y Dividing: y=27(integer, valid)y = 27 \quad \text{(integer, valid)}

Thus, the solution is: N=2106\boxed{N = 2106}


Follow-up Questions

  1. Can you solve a similar problem with different numbers, such as 85 strings and 86 strings?
  2. What does modular arithmetic tell us about the divisors of NN?
  3. How would this change if the hint involved a remainder other than 27?
  4. Can you write a generalized formula for such problems using congruences?
  5. 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

Ask a new question for Free

By Image

Drop file here or Click Here to upload

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