Question

Show that the square of any positive
Integer cannot be of the form 5q +2 or
5q +3 for any integer q.​

Answer 1

Answer:

square of any positive  Integer cannot be of the form 5q +2 or  5q +3 for any integer q.​

Step-by-step explanation:

Let 'a' be any +ve integer and b=5, so by EDL(Euclid's Division Lemma, we can write, a = 5m+r, whre r = 0,1,2,3,4.  There are 5 cases arising

a = 5m, 5m+1,5m+2,5m+3,5m+4 and squaring these 5 cases and by rearranging we will get a^a

= 25m^m = 5 (5m^m) = 5 q                                    (casei)

 = 25m^m+10q+1 = 5(5m^m+2m)+1 = 5q+1           (caseii)

 = 25m^m+20q+4 = 5(5m^m+4m+1)-1 = 5q-1        (caseiii)

= 25m^m+30q+9 = 5(5m^m+3m+1)+4 = 5q+4        (caseiii)

= 25m^m+40q+16 = 5(5m^m+8m+3)+1 = 5q+1        (caseiv)

From all these cases above, we find a square of any +ve integer is not in the form of 5q+2 of 5q+3 for any integer q.