Question

Use Euclid's division algorithm to show that the cube of any positive
integer is of the form 4m or 4m +1 or 4m +3.​

Answer 1

Answer:

let a be the positive integer and b = 4

m is some integer

Euclids division algorithm a = bq + r

a = a, b = 4 , q = q , r = 0, 1, 2 , 3 0<— r<4

if r = 0

a = 4q + 0

a = 4q

cubing on both sides

(a) ³ = (4q) ³

= 64q³

= 4(16³) = 4m

if r = 1

a = 4q + 1

cubing on both sides

(a) ³ = (4q + 1)³

a³ = 64³ + 48q² + 12q + 1

= 4 (16³ + 24q² + 3 ) + 1

= 4m+1

the cube of any positive integer is of the form 4m , 4m + 1 or 4m + 3

Hence proved

HOPE THIS HELPS

@ravalika