Question

show that the square of every positive integer is in the form of 5 p or 5p + 1 or 5 p + 4 where p is any positive integer​

Answer 1

Solution = Let a be the square of any positive integer.

Here a = bp + r

b = 5 , r = 0 ,1 , 2 , 3 , 4 .

Putting the value

1. a = 5p + 0

2. a = 5p + 1

3. a = 5p + 2

4 .a = 5p + 3

5. a = 5p + 4

a = 5p

a = 5p + 1 ( on squaring)

a² = 25p² + 10p + 1

= 5 ( 5p² + 2p ) + 1

= 5p + 1 ( where p = 5p² + 2p )

a² = 5p + 2 ( on Squaring)

a² = 25p² + 20p + 4

= 5 ( 5p² + 4p ) + 4

= 5p + 4 ( where p = 5p² + 4p)

Thus the square of any positive integer be 5p , 5p + 1 , 5p + 4 .

I hope you understand it. Vese to ke question muze bhi kbhi smaj nhi aaya .