Question

Using Euclid’s Division Lemma, show that square of any positive integer is either of the form

4q, 4q+1 for some integer q.​

Answer 1

Answer:

Let positive integer a = 4m + r, By division algorithm, we know here 0 ≤ r < 4 , So

When r = 0

a = 4m  

Squaring both side, we get

a2 = ( 4m )2

a2 = 4 ( 4m​2)

a2 = 4 q , where q = 4m2

When r = 1

a = 4m + 1

squaring both side, we get

a2 = ( 4m + 1)2

a2 = 16m2 + 1 + 8m  

a2 = 4 ( 4m2 + 2m ) + 1  

a2 = 4q + 1 , where q = 4m2 + 2m

When r = 2

a = 4m + 2  

Squaring both hand side , we get

a2 = ​( 4m + 2 )2

a2 = 16m2 + 4 + 16m  

a2 = 4 ( 4m2 + 4m + 1 )

a2 = 4q , Where q = ​ 4m2 + 4m + 1

When r = 3  

a = 4m + 3

Squaring both hand side, we get

a2 = ​( 4m + 3)2

a2 = 16m2 + 9 + 24m  

a2 = 16m2 + 24m ​ + 8 + 1

a2 = 4 ( 4m2 + 6m + 2) + 1

a2 = 4q + 1 , where q = 4m2 + 6m + 2  

Hence,  

Square of any positive integer is in form of 4q or 4q + 1 , where q is any integer.

Answer 2

Answer:

The doctor heard the sound of rats. The sound was a familiar one. He heard this sound four times. The phrases are ‘Again I heard that sound from above’, ‘Again came that noise from above’, ‘Suddenly there came a dull thud as if a rubber tube has fallen’. The sounds stopped after the appearance of the snake.