Originally Posted by
SlipEternal Since you have finite sequences, you can choose any one, and it works. For example, in the first sequence, I can say 24 doesn't belong since every other number listed is the product of at most two primes (but 24 is the product of four primes $\displaystyle 24 = 2^3\cdot 3$).
Alternately, if I consider differences between adjacent terms, I find they are 9-7=2, 13-9=4, 17-13=4, 24-17=7, 35-24=11. Since 35 is the only one that is more than 10 greater than the previous term, I can say that is the one that doesn't belong.
I can create an argument for any term in the sequence. There is no "correct" answer. There are only answer and reason pairs. For each number, I can find a reason why it doesn't belong.
Here are some reasons for the numbers you picked for the next two finite sequences:
1. Neither 250 nor 813 has an element proceeding it.
2. $\displaystyle 706 = 2\cdot 353, 507 = 3\cdot 13^2, 489 = 3\cdot 163, 250 = 2\cdot 5^3$, so 250 is the only number among the four that is the product of more than three primes (not distinct)
$\displaystyle 232 = 2^3\cdot 29, 431 = 431, 612 = 2^2\cdot 3^2\cdot 17, 813 = 3\cdot 271$, so 813 is the only number among the four with two prime factors.
3. $\displaystyle 706-507 = 199, 507-489 = 16, 489-250 = 239$, so 250 is the only one that is more than 200 less than the preceding number.
$\displaystyle 431-232 = 199, 612-431 = 181, 813-612=201$, so 813 is the only one that is more than 200 greater than the preceding number.
The list of reasons why you might choose 250 and 813 respectively is endless.