answer:Using the change of base formula, we can rewrite the equation as [frac{log_a 2}{log_a a} + frac{log_a 3}{log_a a} + frac{log_a 4}{log_a a} = 1.] Simplifying, we get [log_a 2 + log_a 3 + log_a 4 = log_a (2 cdot 3 cdot 4) = log_a 24 = 1.] Since log_a 24 = 1 is equivalent to a^1 = 24, we find that a = boxed{24}. The answer is: 24

answer:The smallest possible sum is 1+1+1=3, and the largest possible sum is 6+6+6=18. There are 18-3+1=16 possible sums. Thus, we need to throw the dice at least boxed{17} times to ensure that the same sum is rolled twice.The answer is: 17

answer:The prime factorization of 24 is 2^3 cdot 3^1. To find the number of positive divisors, we take the exponents of the prime factors and add 1 to each. In this case, there are (3+1)(1+1) = boxed{8} positive divisors of 24. The answer is: 8

answer:If Jason consumes 3 potatoes in 20 minutes, then he consumes 3/20 = 0.15 potatoes per minute. To find out how many minutes it would take for him to eat 27 potatoes, we divide 27 by 0.15: 27/0.15 = 180 minutes. Since there are 60 minutes in an hour, we divide 180 by 60 to find out how many hours it would take: 180/60 = 3 hours. 3 The answer is: 3