Question

Prove that the square of any positive integer is any of the form 5q,5q+1,5q+4 for some positive integer q.

Answer 1

Answer:

Step-by-step explanation:

Eculids division lemma formula is a=bq+r

then r={0,1,2,3,4}, r value must be less than 5

let consider b=5

a=bq+r

let r=0            {(a+b)²=a²+b²+2×a×b)}

a=(5q+0)²

a= (5q)²+(0)²+2×5q×0

a= 25q²+0+0q

a= 25q²

a=5(5q²)........5q³ consider as q

now it will be 5n.

let r=1

a=(5q+1)²

a=(5q)²+(1)²+2×5q×1

a= 25q²+1+11q

a=5[(5q²)+2q]+2.....consider that 5q²+2 as n

now the resulted answer is 5n+2

let r=2

a=(5q+2)²

a=(5q)²+(2)²+2×5q×2

a=25q²+4+20q

a=5[5q²+4q]+4.........consider 5q²+4 as n

now that resulted answer as 5n+4

let r=3

a=(5q+3)²

a=(5q)²+(3)²+2×5q×3

a=25q²+9+30q

a=5[5q²+1+6q]+4........consider 5q²+1+6 as n

now resulted answer is 5n+4

let r=4

a=(5q+4)²

a=(5q)²+(4)²+2×5q×4

a=25q²+16+40q

a=5[5q²+3+8q]+1..........consider 5q²+3+8q as n

now resulted answer is 5n+1

hence we proved that  square of any positive integer is any of the form 5q,5q+1,5q+4 for some positive integer q.