This problem involves solving a Diophantine equation based on the conditions given in the puzzle. Here's how we proceed:

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Problem Setup:

Let $N$ be the total number of coins.

1. From the first condition:
   $N = 77x + 27$, where $x$ is the number of strings (integer).

2. From the second condition:
   $N = 78y$, where $y$ is another integer.

Equating the two expressions for $N$:
$77x + 27 = 78y$

Rearranging terms:
$78y - 77x = 27$

This is a linear Diophantine equation in the form $ax + by = c$, with $a = 78$, $b = -77$, and $c = 27$.

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Step 1: Check for Solvability

A Diophantine equation has integer solutions if and only if the greatest common divisor (GCD) of $a$ and $b$ divides $c$.

• $\text{GCD}(78, 77) = 1$
• Since $1$ divides $27$, the equation is solvable.

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Step 2: Solve Using Extended Euclidean Algorithm

We solve $78y - 77x = 27$ to find a particular solution.

Extended Euclidean Algorithm:

1. Divide 78 by 77:
   $78 = 1 \cdot 77 + 1$
   $77 = 77 \cdot 1 + 0$
   Hence, $\text{GCD}(78, 77) = 1$.

2. Express GCD as a linear combination:
   $1 = 78 - 1 \cdot 77$.

3. Multiply through by 27 to scale up to the desired constant:
   $27 = 27(78) - 27(77)$.

Thus, a particular solution is:
$y_0 = 27$, $x_0 = 27$.

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Step 3: General Solution

The general solution to a Diophantine equation is:
$y = y_0 + k \cdot \frac{b}{\text{GCD}(a, b)}$ $x = x_0 - k \cdot \frac{a}{\text{GCD}(a, b)}$

Here, $\frac{b}{\text{GCD}(a, b)} = -77$ and $\frac{a}{\text{GCD}(a, b)} = 78$:
$y = 27 + k(-77)$ $x = 27 - k(78)$

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Step 4: Find Positive Integer Solutions

We require $N = 78y > 0$ and $x > 0$. From $y = 27 - 77k$, $k$ must ensure $y$ is positive.

Solve for $y > 0$:

$27 - 77k > 0 \implies k < \frac{27}{77}.$ The only valid $k$ is $k = 0$.

Substitute $k = 0$:

$y = 27, \quad x = 27.$

Compute $N$:

$N = 78y = 78(27) = 2106.$

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Final Answer:

The total number of coins is $\mathbf{2106}$.

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Would you like further details or assistance with other problems?

Related Questions:

1. What are the properties of Diophantine equations in general?
2. How does the Extended Euclidean Algorithm work in detail?
3. Can you show more examples of linear Diophantine equations?
4. What happens if the GCD condition is not met in such problems?
5. How does one apply this approach to non-linear equations?

Tip:

Always verify the solvability of a Diophantine equation by checking the GCD condition before