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Problem 434

Algebra Difficulty 2.7 Find the answer Japan Junior Mathematical Olympiad · Japan

Calculate the following number:
877×879−121×123. 877 \times 879 - 121 \times 123.

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

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Official solution

Using (a−b)(a+b)=a2−b2(a-b)(a+b) = a^2-b^2, we obtain
877×879−121×123=(878−1)(878+1)−(122−1)(122+1)=(8782−1)−(1222−1)=8782−1222=(878−122)(878+122)=756×1000=756000. \begin{aligned} 877 \times 879 - 121 \times 123 &= (878 - 1)(878 + 1) - (122 - 1)(122 + 1) \\ &= (878^2 - 1) - (122^2 - 1) = 878^2 - 122^2 \\ &= (878 - 122)(878 + 122) = 756 \times 1000 = 756000. \end{aligned}

Alternatively,
877×879−121×123=877×(1000−121)−121×123=877×1000−121×(877+123)=877×1000−121×1000=756000. \begin{aligned} 877 \times 879 - 121 \times 123 &= 877 \times (1000 - 121) - 121 \times 123 \\ &= 877 \times 1000 - 121 \times (877 + 123) \\ &= 877 \times 1000 - 121 \times 1000 = 756000. \end{aligned}

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