To find the numerical equivalent of \( 102 \cdot 103 \), let's calculate the product:
\[ 102 \cdot 103 = 10200 + 306 = 10506 \]
However, it's more efficient to use the property of numbers close to 100.
We can use the formula for the product of numbers adjacent to a base:
\[ (a+b)(a+c) = a^2 + (b+c)a + bc \]
In this case, let \( a = 100 \), \( b = 2 \), and \( c = 3 \):
\[ (100 + 2)(100 + 3) = 100^2 + (2 + 3) \cdot 100 + 2 \cdot 3 \] \[ = 10000 + 500 + 6 = 10506 \]
Thus, the product \( 102 \cdot 103 = 10506 \).
However, the answer choices seem to imply that the question was aiming for something else, possibly \( 10^5 \) is considered as \( 100,000 \). Given that \( 102 \cdot 103 \) I calculated equals \( 10506 \), this number does not match any of the options provided specifically.
Checking options:
- \( 10 \) is just \( 10 \).
- \( 10^5 \) is \( 100000 \), so while it doesn’t match, this could be misleading (as it indicates an alternative simplification).
- \( 100,000 \) represents \( 10^5 \), and could be a confusion between significant figures or misreading.
- \( 1,100 \) does not match either as it's way lower.
Therefore, the answer does not seem to fit any standard formats of the results I derived: \( 10506 \).
To summarize, the product \( 102 \cdot 103 = 10506 \) but based on the responses, the closest interpretation could derive \( 10^5 \) if they implied a misstatement in expected answer versus actual, or an abstract numeric outcome.
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