Math and Logic Puzzles: Redux

[Farren]

Not sure I can do this formally, but I can at least see that it's true.

Spoiler: Math ...

We'll define

n

as an integer. 10

n

+ 5 would yield the set of all integers with the final digit as a 5.

Part one: showing that the cube is divisible by 125.

(10

n

+ 5) * (10

n

+ 5) * (10

n

+ 5) = ...
(100

n²

+ 100

n

+ 25) * (10

n

+ 5) = ...
(1000

n³

+ 1500

n²

+ 750

n

+ 125)

Each term in that final sum is evenly divisible by 125 (since

n

is an integer); therefore, the entire sum is divisible by 125.

Part two: showing that the cube is odd.

Any integer multiplied by 1000, 1500, or 750 has to be divisible by two. 125 is odd. even + even + even + odd = odd. Therefore, the entire sum is odd.

[Farren]

47.

[Farren]

3.14159.

[Farren]

100388/313771.