Step-by-step explanation
Lets use congruence
For the square of an integer lets use congruence module 3
- If the congruence module 3 is 0, then the congruence of the sqaure is 0² = 0
- If the congruence is 1, then the congruence of the square is 1² = 1
- If the congruence is 2, then the congruence of the square is 2² = 4 = 4-3 = 1 (4 and 1 are equal is Z₃)
Thus, the square of an integer has the form 3k (the congruence is 0) or 3k+1 (the congruence is 1).
For the cube, lets use congruence module 9
- if the congruence module 9 is 0, then the congruence of the cube is 0³ = 0
- if the congruence is 1, then the congruence of the cube is 1³ = 1
- if the congruence is 2, then the congruence of the cube is 2³ = 8
- if the congruence is 3, then the congruence of the cube is 3³ = 27 = 0
- if the congruence is k+3, for certain k, then the congruence of the cube is (k+3)³ = k³+9k²+27k+9 = k³. Hence the congruence of the cube is the same after addding 3 to a number. Thus, the congruence of a number with congruence 4,5,6,7 or 8 module 9 is obtained by computing the congruence module 9 of 1,2 or 3.
With the argument given above, we obtain that the congruence module 9 of the cube of a number is always 0,1 ir 8, thus, the cube of an integer has the form 9k, 9k+1 or 9k+8.
As for the fourth power, we take congruence module 5:
- If the congruence module 5 is 0, then the congruence of the fourth power is 0⁴ = 0
- if the congruence is 1, then the congruence of the foruth power is 1⁴ = 1
- If the congruence is 2, then the congruence of the fourth power is 2⁴ = 16 = 16-5*3 = 1
- If the congruence is 3, then the congruence of the fourth power is 3⁴ = 81 = 81-5*16 = 1
- If the congruence is 4, then the congruence of the fourth power is 4⁴ = 256 = 256-5*51 = 1
In any case, the congruence of the fourth power module 5 is either 0 or 1, as a result, the fourth power has the form 5k or 5k+1.