1. **Stating the problem:** We are given two numbers, 661 and 4779, and their remainders when divided by 7. - 661 leaves a remainder of 3 when divided by 7. - 4779 leaves a remainder of 5 when divided by 7. We need to find the remainders when the following expressions are divided by 7: (i) 4779 + 661 (ii) 4779 - 661 2. **Formula and rules:** When a number $a$ leaves a remainder $r_a$ upon division by $m$, we write: $$a \equiv r_a \pmod{m}$$ For sums and differences: $$a + b \equiv (r_a + r_b) \pmod{m}$$ $$a - b \equiv (r_a - r_b) \pmod{m}$$ If the result is greater than or equal to $m$, subtract $m$ to find the remainder. If the result is negative, add $m$ to find the positive remainder. 3. **Applying to the problem:** - Given: $$661 \equiv 3 \pmod{7}$$ $$4779 \equiv 5 \pmod{7}$$ (i) Sum: $$4779 + 661 \equiv 5 + 3 = 8 \pmod{7}$$ Since $8 \geq 7$, subtract 7: $$8 - 7 = 1$$ So, $$4779 + 661 \equiv 1 \pmod{7}$$ (ii) Difference: $$4779 - 661 \equiv 5 - 3 = 2 \pmod{7}$$ Since $2$ is positive and less than 7, the remainder is 2. 4. **Visual explanation:** Imagine a clock with 7 hours (0 to 6). The number 661 points to 3 on this clock, and 4779 points to 5. - Adding 3 and 5 gives 8, which is 1 hour past 7, so the remainder is 1. - Subtracting 3 from 5 gives 2, so the remainder is 2. **Final answers:** (i) Remainder is $1$ (ii) Remainder is $2$