Find the largest prime divisor of 11! + 12!
To find the largest prime divisor of a number, I want to factor it into prime factors and then pick the largest one. But before I start factoring, I notice that 11! + 12! has a common factor of 11!, since 12! = 12 * 11!. So I can write 11! + 12! as 11! * (1 + 12), and then factor 11! into primes. I know that 11! = 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1, and I can easily identify the prime factors: 11, 7, 5, 3, and 2. The largest prime factor of 11! is 11, but I still need to check if 1 + 12 has a larger prime factor. I can see that 1 + 12 = 13, which is also a prime number. So the largest prime divisor of 11! + 12! is 13. 