### Brain Teasers

# Less Than a Google

Before Google was a company, it was a number. A googol is 10^100: a one followed by a hundred zeros.

How many natural numbers less than a googol are both the square of a square and the cube of a cube?

(If N is an integer then N^2 is a square and N^3 is a cube.)

How many natural numbers less than a googol are both the square of a square and the cube of a cube?

(If N is an integer then N^2 is a square and N^3 is a cube.)

### Answer

599The square of a square is (N^2)^2 = N^(2*2) = N^4. The cube of a cube is (N^3)^3 = N^(3*3) = N^9. The least common multiple between 4 and 9 is 36, so a number that is a square of a square and a cube of a cube is a number of the form N^36.

The 36th root of a googol is

(10^100)^(1/36) = 10^(100/36) = 10^(2.77777...) ~ 599.48

so 599 is the largest integer N such that N^36 < 10^100.

Checking this finds that

599^36 ~ 9.7 x 10^99

600^36 ~ 1.03 x 10^100

Since zero is not a natural number, 1^36 = 1 is the first such number and 599^36 is the last, giving 599 numbers.

Hide Answer Show Answer

## What Next?

View a Similar Brain Teaser...

If you become a registered user you can vote on this brain teaser, keep track of which ones you have seen, and even make your own.

### Solve a Puzzle

## Comments

This teaser is too difficult to consider fun, enjoyment or entertaining. It's pure math and should remain in the Classroom. :lol It would be a great teaser if it was so difficult to enjoy. Tone it down, but still good work.

Difficult for some; easy for others.

I think this is actually one of my easiest teasers since the problem is so straightforward to set up and solve, assuming you know how to set it up. My son is in eighth grade and with a little persistence and thinking, he was able to solve it without help.

I make teasers that I would enjoy--they're not for everyone.

I think this is actually one of my easiest teasers since the problem is so straightforward to set up and solve, assuming you know how to set it up. My son is in eighth grade and with a little persistence and thinking, he was able to solve it without help.

I make teasers that I would enjoy--they're not for everyone.

Nothing else is needed other than this smiley:

I'm not sure why people are complaining. I thought this one was pretty easy -- especially compared to your other recent one (half again as much, or something like that) which I could only follow the explanation after a few tries. This one did not need to check the answer to know I had it, but it was fun to be working with the huge numbers and bring them down to size.

Oh, and thanks for spelling "googol" correctly! I was all set to comment on it when I saw the title of the puzzle.

Oh, and thanks for spelling "googol" correctly! I was all set to comment on it when I saw the title of the puzzle.

Very interesting and fun. Nice job.

Its very good, but I think the proof of the first bit saying it must be N to the power of 36 should be more rigourous.

This is an excellent teaser, accessible to brighter and more persistent maths students. I think a hint would be appreciated by those whose goal is 'fun' rather than pure challenge.

This rocks!!!!!!!!!!!!!

what i mean is that it is very Fun!

I would give full marks for fun and difficult!

I would give full marks for fun and difficult!

To post a comment, please create an account and sign in.

## Follow Braingle!