Numbers Which Are Both Squares and Cubes Must be of Form 7k, 7k+1

Suppose a number  
\[x\]
  is both a square and a cube. Then we can write  
\[x=m^2=n^3\]
. Only numbers of certain forms can be both a square and a cube. The table shows such numbers modulo 7.
\[m, \; n\]
\[m^2\]
\[n^3\]
0
\[0 \equiv 0 \; (mod \; 7)\]
\[0 \equiv 0 \; (mod \; 7)\]
1
\[1 \equiv 1 \; (mod \; 7)\]
\[1 \equiv 1 \; (mod \; 7)\]
2
\[4 \equiv 4 \; (mod \; 7)\]
\[8 \equiv 1 \; (mod \; 7)\]
3
\[9 \equiv 2 \; (mod \; 7)\]
\[27 \equiv 6 \; (mod \; 7)\]
4
\[16 \equiv 2 \; (mod \; 7)\]
\[64 \equiv 1 \; (mod \; 7)\]
5
\[25 \equiv 4 \; (mod \; 7)\]
\[125 \equiv 6 \; (mod \; 7)\]
6
\[36 \equiv 1 \; (mod \; 7)\]
\[216 \equiv 6 \; (mod \; 7)\]
7
\[49 \equiv 0 \; (mod \; 7)\]
\[343 \equiv 0 \; (mod \; 7)\]
8
\[64 \equiv 1 \; (mod \; 7)\]
\[512 \equiv 1 \; (mod \; 7)\]
9
\[81 \equiv 4 \; (mod \; 7)\]
\[729 \equiv 1 \; (mod \; 7)\]
From the table, only numbers which have remainder 0 or 1 on division by 7 can be both a square and a cube, so such numbers must be of the form  
\[7k\]
  or  
\[7k_1\]
.

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