28. Find the smallest prime number that can be represented as the sum of two, three, four, five, and six different prime numbers.
28. It is clear that the sought prime number is not less than $2+3+5+7+11+13=41$. The number 41 is prime, but it cannot be represented as the sum of two prime numbers. Indeed, since one of these two numbers must be the even prime number, i.e., 2, the second number would need to be 39, but it is not prime.

The next prime number in sequence, 43, can be represented as the sum of two primes $(43=2+41)$, but it cannot be represented as the sum of six distinct prime numbers.

For the same reason as 41, the prime numbers 47, 53, and 59 are also ruled out. However, the next prime number, 61, satisfies all the conditions of the problem. Indeed, $61=2+59=$ $=3+5+53=2+5+7+47=3+5+7+17+29=2+3+5+7+$ $+13+31$.