- Use Euclid's Division Lemma to show that the square of any integer i

1.	- Use Euclid's Division Lemma to show that the square of any integer iseither of the form 5m. 5m + 1 or 5m + 4 for some integer m.
Answer» Let x be any integer Then x = 5m or x = 5m+1 or x = 5m+4 for integer x. a. If x = 5m, x2= (5m)2= 25m2= 5(5m2) = 5n (where n = 5m2) b. If x = 5m+1, x2= (5m+1)2= 25m2+10m+1 = 5(5m2+2m)+1 = 5n+1 (where n = 5m2+2m ) c. If x = 5m+4, x2= (5m+4)2= 25m2+40m+16 = 5(5m2+8m+3)+1 = 5n+1 (where n = 5m2+8m+3 ) Thus in each of three cases x2is either of the form 5n or 5n+1 for integer n. 5m to 10km isse ratio mai nikalo

- Use Euclid's Division Lemma to show that the square of any integer iseither of the form 5m. 5m + 1 or 5m + 4 for some integer m.

Answer»

Let x be any integer

Then x = 5m or x = 5m+1 or x = 5m+4 for integer x.

a. If x = 5m, x2= (5m)2= 25m2= 5(5m2) = 5n (where n = 5m2)

b. If x = 5m+1, x2= (5m+1)2= 25m2+10m+1 = 5(5m2+2m)+1 = 5n+1 (where n = 5m2+2m )

c. If x = 5m+4, x2= (5m+4)2= 25m2+40m+16 = 5(5m2+8m+3)+1 = 5n+1 (where n = 5m2+8m+3 )

Thus in each of three cases x2is either of the form 5n or 5n+1 for integer n.

5m to 10km isse ratio mai nikalo