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To multiply the numbers 2146782 and 0422812 using the Karatsuba algorithm, which is a divide and conquer approach, we must first understand the core principle behind this technique. Karatsuba’s algorithm reduces the multiplication of two *n*-digit numbers to at most three multiplications of *n/2*-digit numbers, plus some additions and subtractions. Let the two numbers be: * `X = 2146782` * `Y = 0422812` = `422812` (as leading zeros are not significant in integer representation) We will now apply the Karatsuba algorithm step-by-step. --- Step 1: Equalizing the length of the numbers We need both numbers to be of equal length. `X = 2146782` (length = 7) `Y = 0422812` = `422812` (length = 6) To equalize, prepend one 0 to `Y`: `Y = 0422812` = `0422812` (Now both are 7-digit numbers) So, _____ _____ __________ _____ _______ _________ ________ ________ ___ ________ ______ _______.
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