Question

The square of a positive integer cannot be of the form of (K is some integer)

A}3k+1. B}4k+2 C}5k+1. D}5k+4

Answer 1

Answer : Option B

Square of the number can be of form

1.3k or 3k+1

2.4k or 4k+1

3.5k or 5k+1 or 5k+4

Hence square of number can not be of form 4k+2

Here is the proof


1.

Any positive integer n is of the form 2m or 2m + 1
If n = 2m, then
n2 = 4m2 = 4q, where q = m2
If n = 2m + 1, then
n2=(2m+1)2=4m2+4m+1=4m(m+1)+1=4q+1, where q=m(m+1)
4q or 4q+1 only 4q+2 not possible

2.

Any positive integer in is of the form 3q, 3q +1 or, 3q + 2.
If n = 3q, then
n2=9q2=3(3q2)−3m, where m=3q2
If n = 3q + 1, then
n2=9q2+6q+1=3q(3q+2)+1=3m+1, where m=q(3q+2)
If n = 3q + 2, then
n2=(3q+2)2=9q2+12q+4=3(3q2+4q+1)+1=3m+1, where m=3q2+4q+1.
Hence, n2 is of the form 3m or, 3m + 1 but not of the form 3m + 2.


3.

Let x be any positive integer
Then x = 5q or x = 5q+1 or x = 5q+4 for integer x.
If x = 5q, x² = (5q)² = 25q² = 5(5q²) = 5n (where n = 5q² )
If x = 5q+1, x² = (5q+1)² = 25q²+10q+1 = 5(5q²+2q)+1 = 5n+1 (where n = 5q²+2q )
If x = 5q+4, x² = (5q+4)² = 25q²+40q+16 = 5(5q² + 8q + 3)+ 1 = 5n+1 (where n = 5q²+8q+3 )
∴in each of three cases x² is either of the form 5q or 5q+1 or 5q+4 and for integer q.