1) 1135 1! = 1 3! = 1 x 2 x 3 = 6 5! = 1 x 2 x 3 x 4 x 5 = 120 7! = 1 x 2 x 3 x 4 x 5 x 6 x 7 = 5040 Since the numbers in the code are odd and must be less than 720 (since their product equals 720) the code must consist of the numbers 1, 3, and 5. If there are two 5s in the code, the product of all of the factorials would be at least 120 x 120 = 14400 which is too great. There must be a 5 in the code since there is no combination of 1! and 6! that multiply to 720. The only code that works is 1135, as 6 x 120 = 720 and the 1s have no effect on the product. 2) 184 1 should be the number divided so that the result is unchanged, which is the best case scenario since dividing by any other number would decrease the result. Subtraction is the unused operation since subtracting by any number including 1 would decrease the result, whereas multiplication and addition will always either increase the result or leave it unchanged. This leaves 3 remaining possibilities, although there are multiple ways of writing them due to PEMDAS. 15 and 12 being multiplied and 4 being added: 15 x 12 + 4 / 1 = 184 4 + 15 x 12 / 1 = 184 4 + 15 / 1 x 12 = 184 15 and 4 being multiplied and 12 being added: 15 x 4 + 12 / 1 = 72 12 + 15 x 4 / 1 = 72 12 + 15 / 1 x 4 = 72 4 and 12 being multiplied and 15 being added: 12 x 4 + 15 / 1 = 63 15 + 12 x 4 / 1 = 63 15 + 12 / 1 x 4 = 63 The largest possibility is 184. It’s also the only 3-digit result 3) 3 0.5 x 8 = 4 so there are 4 trees with Osmontium 0.125 x 8 = 1 so to be most efficient she should leave one tree with Osmontium Therefore the smallest number of trees that she needs to swap out the Osmontium is 4 -1 = 3 4) 30 Write a system of equations for the three numbers: (a + b) / 2 = 26 (b + c) / 2 = 11 (c + d) / 2 = 15 When simplified: a + b = 52 b + c = 22 c + d = 30 The answer is equal to (a + d) / 2 so to solve it we must find a + d. The best way to do this is to add the first and third equations and subtract the second equation: (a + b) - (b + c) + (c + d) = 52 - 22 + 30 a + d = 60 and therefore their average is 30