Question

Show that the cube of any positive integer is of the form 4p or 4p+1 or 4p+3 for some integer p​

Answer 1

Answer:

the poosible remainders are 0;1;2

apply them in eqation 4p+r

r=0;;1;2

Answer 2

Let a be any positive integer and b=2

Applying Euclid's division lemma a= 2q+r 0≤ r < 2

possible values of r are 0,1,

when r = 0 a= 2q

when r = 1 a= 2q+1

1) a=2q

• (2q)^3 = 8q^3
• 2*2*2q
• 4(2q)
• 4p where p = 2q

2) a=2q+1

• (2q+1)^3
• 8q^3 +3*4q^2*1 + 3*2q*1+1
• 8q^3+ 24q + 6q +1
• 4(2q^3+ 3q +6q) +1
• 4p+1 where p= 2q^3 +3q +6q