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 $(a^{m})^{n} = a^{m n}$ : $2^{25} + 2^{27}$ .
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.

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.