how many pairs of positive integers satisfy 1/m + 4/n = 1/12 where n is an odd integer less than 60

To solve the problem, we need to determine how many pairs of positive integers \((m, n)\) satisfy the equation:

\[
\frac{1}{m} + \frac{4}{n} = \frac{1}{12}
\]

where \( n \) is an odd integer less than 60.

### Step 1: Manipulate the Equation

First, rearrange the equation:

\[
\frac{1}{m} = \frac{1}{12} - \frac{4}{n}
\]

Next, subtract the fractions:

\[
\frac{1}{m} = \frac{n - 48}{12n}
\]

Thus, \( m \) can be expressed as:

\[
m = \frac{12n}{n - 48}
\]

For \( m \) to be a positive integer, \( \frac{12n}{n - 48} \) must be an integer. This implies that \( n - 48 \) must be a divisor of 12.

### Step 2: Determine Possible Values of \( n \)

Since \( n - 48 \) must divide 12, the possible divisors are:

\[
1, 2, 3, 4, 6, 12
\]

Thus, the corresponding values of \( n \) are:

\[
n = 48 + 1 = 49, \quad n = 48 + 2 = 50, \quad n = 48 + 3 = 51, \quad n = 48 + 4 = 52, \quad n = 48 + 6 = 54, \quad n = 48 + 12 = 60
\]

However, \( n \) must be an odd integer less than 60, so the valid values of \( n \) are:

\[
n = 49, 51
\]

### Step 3: Calculate Corresponding Values of \( m \)

Substitute these values of \( n \) back into the expression for \( m \):

- For \( n = 49 \):
  \[
  m = \frac{12 \times 49}{49 - 48} = \frac{588}{1} = 588
  \]

- For \( n = 51 \):
  \[
  m = \frac{12 \times 51}{51 - 48} = \frac{612}{3} = 204
  \]

### Conclusion

There are 2 pairs of positive integers \((m, n)\) that satisfy the equation where \( n \) is an odd integer less than 60:

\[
(588, 49) \quad \text{and} \quad (204, 51)
\]

Thus, the answer is **2 pairs**.

Would you like any further explanation, or do you have other questions?

---

Here are five follow-up questions to consider:
1. How would the solution change if \( n \) were allowed to be any positive integer under 60?
2. Can you generalize the solution method for different fractions instead of \( \frac{4}{n} \)?
3. What if \( m \) were also required to be less than a certain number, say 1000?
4. How does the problem change if \( n \) must be an even integer less than 60?
5. Can you find a different equation structure that would give more than 2 pairs of solutions?

**Tip:** When solving problems involving divisors, carefully analyze all possible divisor combinations to ensure no solutions are missed.

Explore how to solve the equation 1/m + 4/n = 1/12 where n is an odd integer less than 60. This math problem involves finding pairs of positive integers (m, n) that satisfy the equation through manipulation and analysis of divisors.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation