A positive integer $n>1$ is called beautiful if $n$ can be written in one and only one way as $n=a_1+a_2+\cdots+a_k=a_1 \cdot a_2 \cdots a_k$ for some positive integers $a_1, a_2, \ldots, a_k$, where $k>1$ and $a_1 \geq a_2 \geq \cdots \geq a_k$. (For example 6 is beautiful since $6=3 \cdot 2 \cdot 1=3+2+1$, and this is unique. But 8 is not beautiful since $8=4+2+1+1=4 \cdot 2 \cdot 1 \cdot 1$ as well as $8=2+2+2+1+1=2 \cdot 2 \cdot 2 \cdot 1 \cdot 1$, so uniqueness is lost.) Find the largest beautiful number less than 100.
To determine the largest beautiful number less than 100, we need to understand the conditions under which a number \( n \) can be written uniquely as both a sum and a product of positive integers \( a_1, a_2, \ldots, a_k \) where \( k > 1 \) and \( a_1 \geq a_2 \geq \cdots \geq a_k \).

1. **Understanding the Definition of Beautiful Numbers:**
   - A number \( n \) is beautiful if it can be expressed uniquely as \( n = a_1 + a_2 + \cdots + a_k \) and \( n = a_1 \cdot a_2 \cdot \cdots \cdot a_k \).
   - The uniqueness condition implies that there should be only one way to write \( n \) in this form.

2. **Analyzing the Example:**
   - For example, 6 is beautiful because \( 6 = 3 + 2 + 1 \) and \( 6 = 3 \cdot 2 \cdot 1 \), and this is the only way to write 6 in such a form.

3. **Considering Composite Numbers:**
   - For composite numbers with more than 4 divisors, there are usually multiple ways to write them as a product and sum of integers, which would violate the uniqueness condition.

4. **Focusing on Products of Two Primes:**
   - The solution suggests that numbers which are products of two primes are more likely to be beautiful because they have fewer factorizations, reducing the chances of multiple representations.

5. **Finding the Largest Product of Two Primes Less Than 100:**
   - We need to find the largest number less than 100 that is a product of two primes.
   - Let's list the products of two primes less than 100:
     - \( 2 \times 47 = 94 \)
     - \( 3 \times 31 = 93 \)
     - \( 5 \times 19 = 95 \)
     - \( 7 \times 13 = 91 \)
     - \( 11 \times 7 = 77 \)
     - \( 13 \times 5 = 65 \)
     - \( 17 \times 3 = 51 \)
     - \( 19 \times 2 = 38 \)

6. **Checking the Largest Product:**
   - Among these, the largest product is \( 5 \times 19 = 95 \).

7. **Verifying Uniqueness:**
   - We need to verify that 95 can be written uniquely as a sum and product of integers.
   - \( 95 = 19 + 5 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 \)
   - \( 95 = 19 \times 5 \)
   - Since there is no other way to write 95 as a sum and product of integers that satisfies the conditions, 95 is indeed beautiful.

The final answer is \(\boxed{95}\).