Question

Race # 01
Subjective Assessment
Show that cube of any positive integer is of the form 4m, 4m + 1 or 4m +3, for some integer m.​

Answer 1

Step-by-step explanation: (For this problem you have to know congruency in elementary Number Theory)

Note that any integer k ≡ 0,1,2,3 (mod 4)

If k ≡ 0 (mod 4) , then k³ ≡ 0 x 0 x 0 = 0 (mod 4)

If k ≡ 1 (mod 4) , then k³ ≡ 1 x 1 x 1 = 1 (mod 4)

If k ≡ 2 (mod 4) , then k³ ≡ 2 x 2 x 2 = 8 (mod 4) =  0 (mod 4)

If k ≡ 3 (mod 4) , then k³ ≡ 3 x 3 x 3 = 27 (mod 4) = 3 (mod 4)

Therefore k³ ≡ 0,1 or 3 (mod 4)

∴ Every cube is of the form of 4m , 4m + 1 , 4m + 3 for any integer m .