Please help:)
Establish the following by using division algorithm. 1.the square of any integer is either of the form 3k or 3k+1,k is an integer.
2.for any integer a, a^5 is congruent to a modulo 5

Question

Please help:)
Establish the following by using division algorithm. 1.the square of any integer is either of the form 3k or 3k+1,k is an integer. 
2.for any integer a, a^5 is congruent to a modulo 5

ajprincess · Answer

@hartnn @experimentX please help me:)

sirm3d · Answer

(1) any integer n can be represented by 3k + r, r=0,1,2
compute n^2 for each of these cases

sirm3d · Answer

$$n=3k+r\r=0:n=3k\Rightarrow n^2=9k^2=3(3k^2)\r=1:n=3k+1\Rightarrow n^2=(3k+1)^2=9k^2=6k+1=3(3k^2+2k)+1$$

sirm3d · Answer

$$r=2:n=3k+2\n^2=(3k+2)^2=9k^2+12k+4=9k^2+12k+3+1=3(3k^2+4k+1)+1$$

sirm3d · Answer

let n be any integer
by division algorithm, $$n = 3k + r, \quad 0\leq r<3$$

sirm3d · Answer

as for the 2nd question

by the division algorithm
$$a=5k+r\quad,0\leq r< 5$$

or equivalently
$$a=r \pmod 5, \quad 0\leq r < 5$$
compare $r$ and $r^5$ $\pmod 5$

ajprincess · Answer

Thank u so much @sirm3d