Question

20. Prove that the square of any positive integer is any of the form 59,5q +1,54 + 4 for some
integer​

Answer 1

Answer:

Let positive integer a = 5m+ r , By division algorithm we know here 0 ≤ r < 5 , So

When r = 0

Step-by-step explanation:

a = 5m 

Squaring both side , we get

a2 = ( 5m)2

a2 = 5 ( 5m2)

a2 = 5q, where q = 5m2

When r = 1

a = 5m + 1

squaring both side , we get

a2 = ( 5m + 1)2

a2 = 25m2 + 1 + 10m 

a2 = 5 ( 5m2 + 2m) + 1 

a2 = 5q + 1 , where q = 5m2 + 2m

When r = 2

a = 5m + 2 

Squaring both hand side , we get

a2 = ( 5m + 2)2

a2 = 25m2 + 5 + 20m 

a2 = 5 ( 5m2 + 4m + 5)

a2 = 5q , Where q = 5m2 + 5m + 1

When r = 3 

a = 5m + 3

Squaring both hand side , we get

a2 = ( 5m + 3)2

a2 = 25m2 + 9 + 30m 

a2 = 25m2 + 30m + 10- 1

a2 = 5 ( 5m2 + 6m + 2) - 1

a2 = 5q -1 , where q = 5m2 + 6m + 2 

Hence 

Square of any positive integer is in form of 5q or 5q + 4. , where q is any integer.