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MMM #33: Super divisible

The solution to MMM #32 is up at Blinkdagger. Here we go with #33.

What's the prime factorization of the smallest whole number that is divisible by all integers from 1 up to and including 50?


Now at Blinkdagger: Monday Math Madness #32: The Bigger Plot of Land

The Question

Unfortunately our friends Alice and Bob have unknowingly traded their liberty away and are being fenced inside a rectangular pieces of land on the planet Earth, which is assumed to be a perfect sphere of radius 3950 miles.

Alice and Bob are granted one last choice: To decide which plot of land they want to be fenced in.

Both plots of land are bounded by four fences.

Land 1: Two fences run in an exact north-south direction and the other two run in an exact east–west direction. The north-south fences are exactly 10 miles long and east–west fences are exactly 20 miles long.

Land 2: Similar to Land 1, but the north–south fences are 20 miles long and the east–west fences are 10 miles long.

Can you help Alice and Bob to figure out which plot of land has the greater area?

Prize and submission information at Blinkdagger.

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MMM #31: Winner!

This was a very popular contest. 57 of you sent in your solutions and almost every answer was correct.

Buddha Buck was selected by random.org as the winner of this contest. Here is Buddha's solution:

Problem: Simplify (1/i)^(1/i) - i^i (where i is sqrt(-1))


i*i = -1, -i*i = 1, so 1/i = -i, therefore

(1/i)^(1/i) = (1/i)^(-i)

a^(-b) = (1/a)^b, for all a, b, therefore

(1/i)^(-i) = i^i

So, putting this all together...

(1/i)^(1/i) - i^i
= (1/i)^(-i) - i^i
= i^i - i^i
= 0