Divisibility and Multiples | FIO

Question 7

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.

(i) 4779 + 661

(ii) 4779 - 661

Solution

We can find the remainder of a sum or difference by first finding the sum or difference of the individual remainders, and then finding the remainder of that result.

Step 1 — Understanding the given information

Let's represent the numbers using the division algorithm. This means expressing a number as a multiple of the divisor plus a remainder. We are given that when 661 is divided by 7, the remainder is 3. So, we can write 661 as: $661 = 7 \times q_1 + 3$ Here, $q_1$ is the quotient (the whole number result of the division).

We are also given that when 4779 is divided by 7, the remainder is 5. So, we can write 4779 as: $4779 = 7 \times q_2 + 5$ Here, $q_2$ is the quotient.

Diagram 1

Step 2 — Finding the remainder of 4779 + 661

Let's find the sum of the two numbers. We will substitute their expressions from Step 1. $(4779 + 661) = (7 \times q_2 + 5) + (7 \times q_1 + 3)$ We can rearrange the terms by grouping the multiples of 7 and the remainders. $= (7 \times q_2 + 7 \times q_1) + (5 + 3)$ We can factor out 7 from the first part. $= 7 \times (q_2 + q_1) + 8$ Now, we need to find the remainder of 8 when divided by 7. $8 = 7 \times 1 + 1$ So, we can substitute this back into our expression. $= 7 \times (q_2 + q_1) + (7 \times 1 + 1)$ We can factor out 7 again. $= 7 \times (q_2 + q_1 + 1) + 1$ This shows that when $4779 + 661$ is divided by 7, the remainder is 1.

Visually, we can add the remainders directly. We have a remainder of 5 from 4779 and a remainder of 3 from 661. $5 + 3 = 8$ Now, we find the remainder of 8 when divided by 7. $8 \div 7 \text{ gives a remainder of } 1$

$\boxed{\text{Remainder is 1}}$

Step 3 — Finding the remainder of 4779 - 661

Let's find the difference between the two numbers. We will substitute their expressions from Step 1. $(4779 - 661) = (7 \times q_2 + 5) - (7 \times q_1 + 3)$ We can rearrange the terms by grouping the multiples of 7 and the remainders. $= (7 \times q_2 - 7 \times q_1) + (5 - 3)$ We can factor out 7 from the first part. $= 7 \times (q_2 - q_1) + 2$ This shows that when $4779 - 661$ is divided by 7, the remainder is 2.

Visually, we can subtract the remainders directly. We have a remainder of 5 from 4779 and a remainder of 3 from 661. $5 - 3 = 2$ Since 2 is less than 7, the remainder is simply 2. If the result of subtracting remainders were negative (e.g., $3 - 5 = -2$ ), we would add 7 to it until it becomes a positive number less than 7 (e.g., $-2 + 7 = 5$ ).

$\boxed{\text{Remainder is 2}}$

Answer

(i) The remainder of $4779 + 661$ when divided by 7 is 1. (ii) The remainder of $4779 - 661$ when divided by 7 is 2.

More questions in FIO

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(ii) 4779 - 661

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