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Question When divided by 7, the number 661 leaves a remainder of 3, and 4779 leaves a remainder of 5. Without calculating, can you say what remainders the following expressions will leave when divided by 77? Show the solution both algebraically and visually.
(i) 4779 + 661
(ii) 4779 – 661
Given, 661 = K × 7 + 3, where K = 1, 2, 3, 4, … and, 4779 = K × 7 + 5 Algebraic Method: (i) 4779 + 661 4779 = (682 × 7 ) + 5 Remainder = 5 661 = (94 × 7) + 3 Remainder = 3 ∴ \(\frac{4779+661}{7}=\frac{5+3}{7}=\frac{8}{7}\) = 1 = Remainder (ii) 4779 – 661 ∴ \(\frac{4779-661}{7}=\frac{5-1}{7}=\frac{8}{7}\) = 4 = Remainder Visualization Method: (i) 4779 + 661 = (682 × 7) + 5 + (94 × 7) + 3 = 7 × (682 + 94) + 5 + 3 = 7 × 776 + 8 = Divisible by 7 + \(\frac {8}{7}\) = 1, Remainder (ii) 4779 – 661 = (682 × 7) + 5 – (94 × 7) – 3 = 7 × (682 – 94) + 5 – 3 = 7 × 588 + 2 = Divisible by 7 + 2 = 2, Remainder