Separate The Wheat

Algebra Level 5

Define the following numbers as

\[ \mathbb{A} = \left ( \tfrac{1}{168} + \tfrac{1}{175} + \tfrac{1}{600} \right )\left ( \tfrac{1}{168} + \tfrac{1}{175} - \tfrac{1}{600} \right )\left ( \tfrac{1}{168} + \tfrac{1}{600} - \tfrac{1}{175} \right )\left ( \tfrac{1}{175} + \tfrac{1}{600} - \tfrac{1}{168} \right ) \]

\[ \mathbb{B} = \left ( \tfrac{1}{240} + \tfrac{1}{260} + \tfrac{1}{624} \right )\left ( \tfrac{1}{240} + \tfrac{1}{260} - \tfrac{1}{624} \right )\left ( \tfrac{1}{240} + \tfrac{1}{624} - \tfrac{1}{260} \right )\left ( \tfrac{1}{260} + \tfrac{1}{624} - \tfrac{1}{240} \right ) \]

\[ \mathbb{C} = \left ( \tfrac{1}{432} + \tfrac{1}{540} + \tfrac{1}{720} \right )\left ( \tfrac{1}{432} + \tfrac{1}{540} - \tfrac{1}{720} \right )\left ( \tfrac{1}{432} + \tfrac{1}{720} - \tfrac{1}{540} \right )\left ( \tfrac{1}{540} + \tfrac{1}{720} - \tfrac{1}{432} \right ) \]

Evaluate the digit sum of \( \dfrac{1}{\sqrt{\mathbb{A}}} + \dfrac{1}{\sqrt{\mathbb{B}}} + \dfrac{1}{\sqrt{\mathbb{C}}} \).

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