2. From the natural numbers from 1 to 25, Dasha chose six such that the difference between any two chosen numbers is divisible by 4. What is the maximum number of prime numbers that Dasha could have chosen?
Answer: Five. First solution. The difference of two numbers is divisible by 4 if and only if these numbers have the same remainder when divided by 4. Let's list all prime numbers less than 25 and their remainders when divided by 4: 2-2, 3-3, 5-1, 7-3, 11-3, 13-1, 17-1, 19-3, 23-3. The most common remainder is 3, which appears five times, giving us the answer. Second solution. Since $5-2=3,13-11=19-17=2$, from the groups of prime numbers $\{2,3,5\},\{11,13\},\{17,19\}$ Dasha could choose no more than one number from each. Among the primes less than 25, the only numbers not in these groups are 7 and 23, so the number of prime numbers Dasha could choose is no more than five. An example where there are exactly five is given in the first solution.