Factor some more

Number Theory Level 3

191 902 = 438 ⋅ 439 − 19 ⋅ 20. \large 191\:902 = 438\cdot 439 - 19\cdot 20.

Based on this fact, find two factors x , y x, y such that 191 902 = x ⋅ y x > y > 400 191\:902 = x\cdot y \\ x > y > 400 Submit your answer by concatenating x x and y y . For instance, if x = 323 x = 323 and y = 176 y = 176 , submit "323176".


The answer is 458419.

Arjen Vreugdenhil by Arjen Vreugdenhil , 44, USA

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1 solution

191 902 = 438 ⋅ 439 − 19 ⋅ 20 = ( ( 438 1 2 ) 2 − 1 4 ) − ( ( 19 1 2 ) 2 − 1 4 ) = ( 438 1 2 ) 2 − ( 19 1 2 ) 2 = ( 438 1 2 + 19 1 2 ) ⋅ ( 438 1 2 − 19 1 2 ) = 458 ⋅ 419 . 191\:902 = 438\cdot 439 - 19\cdot 20 \\ = \left((438\tfrac12)^2-\tfrac14\right) - \left((19\tfrac12)^2-\tfrac14\right) \\ = (438\tfrac12)^2 - (19\tfrac12)^2 \\ = (438\tfrac12 + 19\tfrac12)\cdot (438\tfrac12 - 19\tfrac12) \\ = \boxed{458\cdot 419}.

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