Multiply 106 × 94 as a difference of squares

Question

The two factors are symmetric about 100, each sits 6 away. That symmetry collapses the product into one subtraction. Solve in your head. What is 106 	imes 94? Aim: under 10 seconds.

Accepted Answer

Both are 6 from the midpoint 100, so use (m+d)(m-d) = m^2 - d^2: 106 	imes 94 = 100^2 - 6^2 = 10000 - 36 = 9964. So 106 	imes 94 = \mathbf{9964}. The whole computation is "ten thousand minus thirty-six." Spotting the midpoint is the trick: (106 + 94)/2 = 100, and the gap to each is 6. Takeaway. When two numbers straddle a round midpoint by the same distance, the product is 	ext{midpoint}^2 - 	ext{gap}^2. Difference of squares turns a multiplication into a subtraction.

Multiply 106 × 94 as a difference of squares

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