Question

Which `3` digit number is both a square and a cube?

Answer 1

`729`

Explanation:

Let `n` be this number.

Then `n = m^6 = (m^2)^3 = (m^3)^2` so a candidate is `m = 3` because

`n = 729 = (3^2)^3 = (3^3)^2`

Answer 2

`(3^2)^3 =9^3 = 729`

`(3^3)^2 = 27^2 = 729`

Explanation:

One way to approach this is to consider the cubes and see which meet the other conditions.

`1^3 = 1^2 = 1 = 1^6` but it is a one digit number
`2^3 = 8`
`3^3 = 27`
`4^3 = 64 = 8^2 = 2^6`- a two digit number
`5^3 = 125`
`6^3 = 216`
`7^3 = 343`
`8^3 = 512`
`9^3 = 729 = 27^2 = 3^6`
`10^3 = 1000`

There are 3 cubes which are also square numbers, but only `729` is a three digit number. The next would be `4^6 =4096= 16^3 = 64^2` a four-digit number.

Note that `1,4 and 9` are the first three square numbers,
Their cubes are therefore both squares and cubes.
We could also have started with integers which are cubes, and then square them:

The cubes are:`1,8,27, 64 ...`

`1^2 = 1`
`8^2=64`
`27^2 =729" "larr` a three-digit number
`64^2 =4096`

However, if we look at the indices:

`(x^2)^3 = (x^3)^2 = x^6`

Find the numbers which are sixth powers and have `3` digits:

`1^6 = 1`
`2^6 = 64`
`3^6 = 729`
`4^6 = 4096`