(a) What is the sum of the digits of the smallest positive integer which is divisible by 99 and has all of its digits equal to 2? (b) When 270 is divided by the odd number n, the quotient is a prime number and the remainder is 0. What is n?

For part (a), a number made only of 2s is divisible by 99 only if it's divisible by both 9 and 11; divisibility by 9 requires the digit sum, which is $$2$$ times the number of digits, to be a multiple of 9, and checking against divisibility by 11 shows 18 twos is too few but 18 twos actually fails 11's alternating test, so the correct count of digits making the sum a multiple of 9 and satisfying divisibility by 11 gives a digit sum of 36. For part (b), $$\frac{270}{n}$$ must be an odd prime's complement, which forces the prime quotient to be 2 (the only even prime) so that n stays odd, giving $$n = \frac{270}{2} = 135$$.