32⁵ + 2²⁷ is divisible by

Correct Option (C)

The given expression is ${32}^5+2^{27}$.

• First, express 32 as a power of 2: $32=2^5$.
• Substitute this into the expression: $(2^5{)}^5+2^{27}$.
• Apply the exponent rule $(a^m{)}^n=a^{mn}$: $2^{25}+2^{27}$.
• Factor out the common term $2^{25}$: $2^{25}(1+2^2)$.
• Simplify the term in the parenthesis: $2^{25}(1+4)=2^{25}\times 5$.
• To demonstrate divisibility by 10, rewrite $2^{25}$ as $2^{24}\times 2^1$: $2^{24}\times 2\times 5$.
• This simplifies to $2^{24}\times 10$.

Since the expression can be written as $2^{24}\times 10$, it is a multiple of 10 and therefore divisible by 10.

Incorrect Options:

The simplified form of the expression is $2^{25}\times 5$. This product consists solely of prime factors 2 and 5.

• Option A (3): The prime factorization of the expression does not include 3. Therefore, the expression is not divisible by 3.
• Option B (7): The prime factorization of the expression does not include 7. Therefore, the expression is not divisible by 7.
• Option D (11): The prime factorization of the expression does not include 11. Therefore, the expression is not divisible by 11.