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question:f the sum of the factorials 1! + 2! + 3! + cdots + 49! + 50! is divided by 15, what is the remainder?
answer:Note that the factorial of any number greater than or equal to 5 is divisible by 15, since it will have a factor of 5 and a factor of 3. So, we only need to consider the sum of the factorials from 1! to 4!, which is 1+2+6+24=33. Dividing 33 by 15 gives a quotient of 2 and a remainder of boxed{3}. The answer is: 3
question:f Olga has an aquarium with fish in 5 different colors, including 12 yellow ones, half as many blue ones, and twice as many green ones as yellow ones, what is the total number of fish in her aquarium?
answer:Olga has 12 yellow fish in her aquarium. She has half as many blue fish as yellow fish, so she has 12/2 = 6 blue fish. She has twice as many green fish as yellow fish, so she has 12*2 = 24 green fish. In total, Olga has 12 yellow + 6 blue + 24 green = 42 fish in her aquarium. 42 The answer is: 42
question:A triangle has sides of length 5 and 6 units. The length of the third side is x units, where x is an integer. What is the largest possible perimeter of the triangle?
answer:By the triangle inequality, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In this case, we have the side lengths 5 and 6 units. To maximize the perimeter, we want the third side to be as long as possible. So, the third side must satisfy 5+6>x. The largest possible value of x is 5+6-1=10. Therefore, the largest possible perimeter is 5+6+10=boxed{21} units. The answer is: 21
question:f Madeline wants to plant 20 flowers in her garden and expects that half of the seeds she plants will not bloom, how much money will she need to spend on seed packs if each pack contains 25 seeds and costs 5?
answer:Madeline wants to plant 20 flowers, so she needs to buy 20 x 2 = 40 seeds. Each seed pack contains 25 seeds, so she needs to buy 40 / 25 = 1.6 packs of seeds. Since she can't buy a fraction of a pack, she will need to buy 2 packs of seeds. Each pack of seeds costs 5, so she will need to spend 2 x 5 = 10 on seed packs. 10 The answer is: 10