Given that \( B \) is 10 times larger than \( A \), we can express this relationship mathematically as:
\[ B = 10A \]
Now, we know that the number 10 can be factored into its prime factors as follows:
\[ 10 = 2 \times 5 \]
Thus, we can express \( B \) in terms of \( A \):
\[ B = 10A = (2 \times 5)A \]
Now, substituting \( B = 2^x \times 5 \) into the equation we have for \( B \):
\[ 2^x \times 5 = (2 \times 5)A \]
This can be rewritten as:
\[ 2^x \times 5 = 2^1 \times 5^1 \times A \]
To isolate \( A \), we can divide both sides of the equation by \( 2 \times 5 \):
\[ A = \frac{2^x \times 5}{2^1 \times 5^1} = 2^{x-1} \times 5^{1-1} = 2^{x-1} \times 5^0 \]
Since \( 5^0 = 1 \), we can simplify this to:
\[ A = 2^{x-1} \times 1 = 2^{x-1} \]
So the prime factorization of \( A \) is:
\[ \boxed{2^{x-1}} \]
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