### 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.

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