Question

Which of the following numbers is divisible by \( 3^{10} + 2 \)?

• A. \( 3^{30} + 2 \)
• B. \( 3^{20} + 4 \)
• C. \( 3^{30} + 8 \)
• D. \( 3^{20} + 8 \)

Hint:

Remember the factorization formulas for sums and differences of powers, especially for cubes: \( a^3 + b^3 = (a+b)(a^2-ab+b^2) \) and \( a^3 - b^3 = (a-b)(a^2+ab+b^2) \). These are very useful in divisibility problems.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem tests the knowledge of algebraic factorization, specifically the sum of cubes identity. We need to check which of the given options can be expressed as a product that includes \( (3^{10} + 2) \) as a factor.
Step 2: Key Formula or Approach:
The key algebraic identity is the sum of cubes: \[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \] We will try to apply this identity to the options by letting \( a = 3^{10} \).
Step 3: Detailed Explanation:
Let \( x = 3^{10} \). The question asks which of the options is divisible by \( x + 2 \). Let's examine each option by substituting \( x = 3^{10} \):

\( 3^{30} + 2 = (3^{10})^3 + 2 = x^3 + 2 \). This is not directly divisible by \(x+2\). (According to the Factor Theorem, for \(x^3+2\) to be divisible by \(x+2\), \(x=-2\) should be a root, i.e., \( (-2)^3+2 = -8+2 = -6 \neq 0 \)).
\( 3^{20} + 4 = (3^{10})^2 + 4 = x^2 + 4 \). This is a sum of squares, which does not generally factor into \( (x+2) \).
\( 3^{30} + 8 = (3^{10})^3 + 8 = x^3 + 8 \). We can write 8 as \( 2^3 \). So the expression is \( x^3 + 2^3 \). This is a sum of cubes. We can use the identity \( a^3 + b^3 = (a+b)(a^2 - ab + b^2) \) with \( a = x \) and \( b = 2 \). \[ x^3 + 2^3 = (x + 2)(x^2 - 2x + 2^2) = (x + 2)(x^2 - 2x + 4) \] Substituting back \( x = 3^{10} \): \[ 3^{30} + 8 = (3^{10} + 2)((3^{10})^2 - 2(3^{10}) + 4) = (3^{10} + 2)(3^{20} - 2 \cdot 3^{10} + 4) \] Since \( 3^{30} + 8 \) can be written as a product of \( (3^{10} + 2) \) and another integer, it is divisible by \( 3^{10} + 2 \).
\( 3^{20} + 8 = (3^{10})^2 + 8 = x^2 + 8 \). This does not factor into \( (x+2) \).
Step 4: Final Answer:
The number \( 3^{30} + 8 \) is divisible by \( 3^{10} + 2 \).