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'Equal Length Powers' printed from https://nrich.maths.org/
Answer: 3 (1, 2 and 4)
Listing up from 1
$n$ |
$n^2$ |
$n^3$ |
Same number
of digits? |
1 |
1 |
1 |
yes |
2 |
4 |
8 |
yes |
3 |
9 |
27 |
no |
4 |
16 |
64 |
yes |
5 |
25 |
125 |
no |
... |
2-digit |
3-digit |
no |
10 |
100 |
1000 |
no |
... |
3-digit |
4-digit |
no |
When $n\gt$10, multiplying by $n$ will increase the number of digits, so $n^3$ (which is $n^2\times n$) will have more digits than $n^2.$
Counting down from 10
When $n\ge$10, multiplying by $n$ will increase the number of digits, so $n^3$ (which is $n^2\times n$) will have more digits than $n^2.$
Numbers less than 10 have squares less than 100 (1 or 2 digits).
Cubes less than 100: 4$^3$ = 64, 5$^3$ = 125
1, 2, 3 and 4 have cubes less than 100
1, 2, 3 have 1-digit squares but 4$2$ = 16 also has 2 digits.
Cubes less than 10: 2$^3$ = 8, 3$^3$ = 27
1 and 2 have cubes and squares less than 10 (1 digit)
So 1, 2 and 4 are the only possibilites.