Find a​ 3-by-3 magic square using the numbers 4​, 5​, 6​, 14​, 15​, 16​, 24​, 25​, and 26.

A monkey has filled in a 3 × 3 grid with the numbers 1, 2, . . . , 9. A cat writes down thethree numbers obtained by multiplying the numbers in each horizontal row. A dog writesdown the three numbers obtained by multiplying the numbers in each vertical column.Can the monkey fill in the grid in such a way that the cat and dog obtain the same listsof three numbers? What if the monkey writes the numbers 1, 2, . . . , 25 in a 5×5 grid? Or1, 2, . . . , 121 in a 11 × 11 grid? Can you find any conditions on n that guarantee that itis possible or any conditions that guarantee that it is impossible for the monkey to writethe numbers 1, 2, . . . , n^2 in an n × n grid so that the cat and the dog obtain the same lists of numbers? Please provide a mathematically rigorous solution with proofs that could be understood by a person at a high school level.

Ellie is putting together a balloon bouquet for her mom's birthday party. She wants to fill the bouquet with 48 balloons, and wants to make sure that the colors are perfectly divided into equal groups so that these are the same amount of every color. How many different ways can Ellie divide the balloons into equal groups such that each group contains the same number of balloons? The answer will require finding all the factors of 48.

Find the highest power of 17 that divides 1500!