Let a, b, and c be different positive integers such that 1 = 1/2 + 1/3 + 1/7 + 1/a + 1/b + 1/c. What is the smallest possible value of a + b + c?

We are given the equation:

\[
1 = \frac{1}{2} + \frac{1}{3} + \frac{1}{7} + \frac{1}{a} + \frac{1}{b} + \frac{1}{c}
\]

where \(a\), \(b\), and \(c\) are distinct positive integers, and we need to find the smallest possible value of \(a + b + c\).

### Step 1: Simplify the left-hand side

First, calculate the sum of the known fractions:

\[
\frac{1}{2} + \frac{1}{3} + \frac{1}{7}
\]

To add these fractions, we need a common denominator. The least common denominator (LCD) of 2, 3, and 7 is \(42\). Rewrite the fractions with denominator 42:

\[
\frac{1}{2} = \frac{21}{42}, \quad \frac{1}{3} = \frac{14}{42}, \quad \frac{1}{7} = \frac{6}{42}
\]

Now add them:

\[
\frac{21}{42} + \frac{14}{42} + \frac{6}{42} = \frac{41}{42}
\]

Thus, we have:

\[
1 = \frac{41}{42} + \frac{1}{a} + \frac{1}{b} + \frac{1}{c}
\]

### Step 2: Simplify the equation

Subtract \(\frac{41}{42}\) from both sides:

\[
\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1 - \frac{41}{42} = \frac{1}{42}
\]

Thus, we now need to find distinct positive integers \(a\), \(b\), and \(c\) such that:

\[
\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{42}
\]

### Step 3: Find suitable values for \(a\), \(b\), and \(c\)

Let's try different values for \(a\), \(b\), and \(c\) by testing values of \(a\), solving for \(b\) and \(c\), and ensuring that they are distinct positive integers.

#### Case 1: \(a = 43\)

Substituting \(a = 43\) into the equation, we get:

\[
\frac{1}{b} + \frac{1}{c} = \frac{1}{42} - \frac{1}{43}
\]

Find the common denominator of 42 and 43:

\[
\frac{1}{42} - \frac{1}{43} = \frac{43 - 42}{42 \times 43} = \frac{1}{1806}
\]

So, we need \(b\) and \(c\) such that:

\[
\frac{1}{b} + \frac{1}{c} = \frac{1}{1806}
\]

Try \(b = 1807\) and \(c = 1806\):

\[
\frac{1}{1807} + \frac{1}{1806} = \frac{1806 + 1807}{1806 \times 1807} = \frac{3613}{1806 \times 1807}
\]

which is clearly not equal to \(\frac{1}{1806}\).

Thus, \(b = 

Explore how to solve a number theory problem involving fractions, where a, b, and c are distinct integers. This detailed solution covers simplifying fractions using common denominators and solving equations. Suitable for advanced students in Grades 10-12.

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