Choose any 3 numbers. When is their sum divisible by 3? Explore all possible cases and generalise.

Question

Homi · Accepted Answer

The sum of three numbers is divisible by 3 if the sum of their remainders when divided by 3 is also divisible by 3.

Step 1 — Representing Numbers with Remainders

Let us understand how numbers behave when divided by 3. Any whole number can be written in one of three forms. These forms are $3k$ , $3k+1$ , or $3k+2$ . Here, $k$ is a whole number. $3k$ means the remainder is 0. $3k+1$ means the remainder is 1. $3k+2$ means the remainder is 2. Let our three numbers be $A$ , $B$ , and $C$ . We can write them using their remainders. $A = 3k_1 + r_1$ $B = 3k_2 + r_2$ $C = 3k_3 + r_3$ Here, $r_1$ , $r_2$ , $r_3$ are their remainders. Each $r_i$ can be 0, 1, or 2. Let us find the sum of these three numbers. $A + B + C = (3k_1 + r_1) + (3k_2 + r_2) + (3k_3 + r_3)$ $= 3k_1 + 3k_2 + 3k_3 + r_1 + r_2 + r_3$ $= 3(k_1 + k_2 + k_3) + (r_1 + r_2 + r_3)$ Let $K = k_1 + k_2 + k_3$ . This $K$ is also a whole number. The term $3K$ is always divisible by 3. The sum $A+B+C$ is divisible by 3. This happens if $r_1+r_2+r_3$ is divisible by 3.

$\boxed{\text{Sum of numbers is divisible by 3 if sum of remainders is divisible by 3.}}$

Diagram 1

Step 2 — Possible Sums of Remainders

We need to find when $r_1 + r_2 + r_3$ is divisible by 3. Each remainder $r_i$ can be 0, 1, or 2. Let us find the smallest possible sum of remainders. $0+0+0 = \mathbf{0}$ Let us find the largest possible sum of remainders. $2+2+2 = \mathbf{6}$ So, the sum of remainders can be 0, 1, 2, 3, 4, 5, or 6. We are looking for sums that are divisible by 3. These sums are 0, 3, and 6.

$\boxed{\text{The sum of remainders must be 0, 3, or 6.}}$

Step 3 — Exploring Cases for Divisibility

Let us explore the cases where the sum of remainders is 0, 3, or 6.

Case 1: Sum of remainders is 0. This happens only if all three remainders are 0. So, $r_1=0$ , $r_2=0$ , $r_3=0$ . This means all three numbers are divisible by 3. Let us choose three numbers: 3, 6, 9. Their remainders are: $3 \div 3 \implies \text{remainder } \mathbf{0}$ $6 \div 3 \implies \text{remainder } \mathbf{0}$ $9 \div 3 \implies \text{remainder } \mathbf{0}$ The sum of remainders is: $0+0+0 = \mathbf{0}$ The sum of the numbers is: $3+6+9 = \mathbf{18}$ Since 18 is divisible by 3, this case works.

Case 2: Sum of remainders is 3. This can happen in two ways. (a) One remainder is 0, one is 1, and one is 2. For example, $r_1=0, r_2=1, r_3=2$ . Let us choose three numbers: 3, 4, 5. Their remainders are: $3 \div 3 \implies \text{remainder } \mathbf{0}$ $4 \div 3 \implies \text{remainder } \mathbf{1}$ $5 \div 3 \implies \text{remainder } \mathbf{2}$ The sum of remainders is: $0+1+2 = \mathbf{3}$ The sum of the numbers is: $3+4+5 = \mathbf{12}$ Since 12 is divisible by 3, this case works.

(b) All three remainders are 1. For example, $r_1=1, r_2=1, r_3=1$ . Let us choose three numbers: 1, 4, 7. Their remainders are: $1 \div 3 \implies \text{remainder } \mathbf{1}$ $4 \div 3 \implies \text{remainder } \mathbf{1}$ $7 \div 3 \implies \text{remainder } \mathbf{1}$ The sum of remainders is: $1+1+1 = \mathbf{3}$ The sum of the numbers is: $1+4+7 = \mathbf{12}$ Since 12 is divisible by 3, this case works.

Case 3: Sum of remainders is 6. This happens only if all three remainders are 2. So, $r_1=2$ , $r_2=2$ , $r_3=2$ . This means all three numbers have a remainder of 2. Let us choose three numbers: 2, 5, 8. Their remainders are: $2 \div 3 \implies \text{remainder } \mathbf{2}$ $5 \div 3 \implies \text{remainder } \mathbf{2}$ $8 \div 3 \implies \text{remainder } \mathbf{2}$ The sum of remainders is: $2+2+2 = \mathbf{6}$ The sum of the numbers is: $2+5+8 = \mathbf{15}$ Since 15 is divisible by 3, this case works.

$\boxed{\text{The sum of three numbers is divisible by 3 in these cases.}}$

Step 4 — Generalisation

We have explored all possible combinations of remainders. The sum of three numbers is divisible by 3 in specific situations. This happens when the sum of their individual remainders is divisible by 3. This occurs in two main scenarios. Scenario 1: All three numbers have the same remainder when divided by 3. This means all three remainders are 0, or all are 1, or all are 2. Scenario 2: The three numbers have different remainders when divided by 3. This means one number has remainder 0, one has remainder 1, and one has remainder 2.

Answer

The sum of three numbers is divisible by 3 if: (i) All three numbers have the same remainder when divided by 3 (all 0, all 1, or all 2). (ii) The three numbers have different remainders when divided by 3 (one 0, one 1, and one 2).

Question 29

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