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 7? Show the solution both algebraically and visually.
4779 + 661
The remainder when (4779 + 661) is divided by 7 is 1.
4779 - 661
The remainder when (4779 - 661) is divided by 7 is 2.
Explanation
(i) Algebraically: 4779 ≡ 5 (mod 7) and 661 ≡ 3 (mod 7). Therefore, 4779 + 661 ≡ 5 + 3 = 8 ≡ 1 (mod 7). Visually: If we think of 4779 as 7 groups plus 5 extra, and 661 as 7 groups plus 3 extra, combining them gives 7 groups plus 8 extra, which can form one more complete group of 7, leaving 1 extra. (ii) Algebraically: 4779 ≡ 5 (mod 7) and 661 ≡ 3 (mod 7). Therefore, 4779 - 661 ≡ 5 - 3 = 2 (mod 7). Visually: If 4779 has 5 extra over groups of 7, and 661 has 3 extra, subtracting means removing those 3 extra from the 5 extra, leaving 2 extra. The result is 2 more than a multiple of 7.