3. 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)

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 \( N \), which satisfies:

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

Where \( x \) and \( y \) are integers.

### Step 1: Combine the equations
From Equation 1 and Equation 2, we equate the expressions for \( N \):
\[
77x + 27 = 78y
\]

Rearranging gives:
\[
78y - 77x = 27
\]

### Step 2: Solve using modular arithmetic
Rewriting the equation:
\[
78y \equiv 27 \pmod{77}
\]

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

Thus, \( y = 27 + 77k \), where \( k \) is an integer.

### Step 3: Find \( N \)
Substituting \( y = 27 + 77k \) into Equation 2:
\[
N = 78y = 78(27 + 77k)
\]
\[
N = 78 \cdot 27 + 78 \cdot 77k
\]
\[
N = 2106 + 6006k
\]

### Step 4: Verify minimal \( N \)
For \( k = 0 \), \( N = 2106 \). Verifying this:
1. If \( N = 77x + 27 \):
   \[
   2106 = 77x + 27
   \]
   Subtract 27:
   \[
   2079 = 77x
   \]
   Dividing:
   \[
   x = 27 \quad \text{(integer, valid)}
   \]

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

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

---

### 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 \( N \)?
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

This is an ancient puzzle problem: 'Yen Kung, 1372. We have an 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?'

Learn how to solve the ancient Yen Kung puzzle from 1372, which involves modular arithmetic and simultaneous congruences. Discover the number of coins required when 77 strings leave you 50 coins short, but 78 strings are exact. A detailed solution using key number theory concepts is provided. Suitable for Grades 10-12.

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