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The problem asks us to find the largest "beautiful" number less than 100, based on the given definition. Let me break down the concept and solution step-by-step:

### Problem Summary:
A positive integer \( n > 1 \) is called "beautiful" if it can be written in **one and only one way** as a product of integers:
\[
n = a_1 + a_2 + \cdots + a_k = a_1 \cdot a_2 \cdot \cdots \cdot a_k
\]
where \( a_1 \leq a_2 \leq \cdots \leq a_k \) are positive integers.

- **Example**: \( 6 = 3 + 2 + 1 = 3 \times 2 \times 1 \) is beautiful because there's only one way to express 6 in this form.
- **Counterexample**: \( 8 = 4 + 2 + 1 = 4 \times 2 \times 1 \) and \( 8 = 2 + 2 + 2 + 2 = 2 \times 2 \times 2 \times 2 \), which gives multiple expressions, so 8 is not beautiful.

### Objective:
Find the largest beautiful number less than 100.

### Approach:
We need to systematically check numbers up to 100 to determine if each number has a **unique** factorization where the sum and product of the factors are equal. Let me calculate the beautiful numbers and determine the largest one below 100.

I'll proceed with solving this.It seems that the initial approach was too slow to compute all possibilities efficiently. I'll refine the method to optimize the search for beautiful numbers by focusing on more feasible combinations and shortcuts to find the largest beautiful number under 100. Let me attempt a more efficient solution.

A positive integer n > 1 is called beautiful if it can be written in one and only one way as n = a1 + a2 + ... + ak = a1 * a2 * ... * ak for some positive integers a1, a2, ..., ak. Find the largest beautiful number less than 100.

Solve the problem of finding the largest beautiful number less than 100, based on unique factorizations where the sum and product of the factors are equal. This number theory problem is perfect for high school students interested in combinatorics and factorization.

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