Which term in the following is perfect divisible by $$(x^8 + 1)?$$​​

Correct option is C

Concept Used:

If a number like $$x^8 + 1$$​ divides another expression like $$x^k + 1,$$​ then:

The exponent k should be a multiple of 8,

and the quotient $$\frac{k}{8}$$​ should be odd.

In simple terms:

→ The power in the expression should be divisible by 8,

→ And after dividing by 8, the result should be an odd number.

Solution:

(A) $$x^{36} + 1:$$​​

$$36 \div 8$$​= 4.5 → Not a whole number → Not divisible

(B) $$x^{16} + 1:$$​​

$$16 \div 8$$​ = 2 → Even → Not divisible

(C) $$x^{72} + 1:$$​​

$$72 \div 8$$​ = 9 → Odd → Divisible

(D) $$x^{64} + 1:$$​​
$$64 \div 8$$​ = 8 → Even → Not divisible

Thus, Option (C) $$x^{72} + 1$$​ is perfectly divisible by $$x^8 + 1$$​​