Divisibility Rule for 3 (original) (raw)

Last Updated : 13 Sep, 2025

**The divisibility rule of 3 is a simple mathematical guideline used to determine whether a given integer is divisible by 3 without performing the actual division operation.

This rule simplifies the process of checking whether a number is divisible by 3. By this, we can quickly determine the divisibility without performing long division. This rule is particularly useful in mental math, simplifying fractions, and various mathematical applications.

Divisibility Rule of 3 Proof

Any number N can be expressed as a sum of its digits multiplied by powers of 10. For example, the number 5432 can be written as:

N = 5 \cdot 10^3 + 4 \cdot 10^2 + 3 \cdot 10^1 + 2 \cdot 10^0

In general, for a number with digits d_n, d_{n-1}, \dots, d_1, d_0, it can be expressed as:

N = d_n \cdot 10^n + d_{n-1} \cdot 10^{n-1} + \dots + d_1 \cdot 10^1 + d_0 \cdot 10^0

Next, let's look at powers of 10 modulo 3:

• 10^0 = 1 \equiv 1 ~(\mod 3)
• 10^1 = 10 \equiv 1 ~(\mod 3)
• 10^2 = 100 \equiv 1~ (\mod 3)
• 10^3 = 1000 \equiv 1~ (\mod 3)

In fact, for all n ≥ 1, 10n ≡ 1 mod 3. This means that each power of 10, when divided by 3, gives a remainder of 1.

Since 10n ≡ 1 mod 3, the expression for N becomes:

N \equiv d_n \cdot 1 + d_{n-1} \cdot 1 + \dots + d_1 \cdot 1 + d_0 \cdot 1 ~(\mod 3)

N \equiv d_n + d_{n-1} + \dots + d_1 + d_0 ~(\mod 3)

This means that the remainder when dividing N by 3 is the same as the remainder when dividing the **sum of its digits by 3.

Thus, for a number N to be divisible by 3, the sum of its digits must be divisible by 3. If the sum of the digits of N is divisible by 3, then the entire number N is divisible by 3.

Verification with Table of 3

The following are the numbers in the table of 3 and their sum of digits. We can clearly see that all digit sums are multiples of 3

Divisibility Rule of 3 for Large Numbers

The divisibility rule of 3 for large numbers follows the same principle as for smaller numbers. The divisibility rule of 3 for large numbers is an arithmetic shortcut that helps determine whether a given large number is divisible by 3 without performing the actual division.

**Procedure for Large Numbers:

• **Sum the Digits: Add up all the digits of the given number.
• **Check the Sum: Determine if the sum obtained in step 1 is divisible by 3. (You can keep summing up the digits if the resultant sum is large.)
• **Conclusion: If the sum is divisible by 3, then the original number is also divisible by 3; otherwise, it is not.

**Example: Consider the large number 12394567891239456789: Check its divisibility by 3 without performing actual division.

**Solution:

• **Sum the Digits: 1 + 2 + 3 + 9 + 4 + 5 + 6 + 7 + 8 + 9 + 1 + 2 + 3 + 9 + 4 + 5 + 6 + 7 + 8 + 9 = 108
• **Sum up the digits again : 1 + 0 + 8 = 9
• **Check the Sum: 9 is divisible by 3.
• **Conclusion: Therefore, the large number 123456789123456789 is divisible by 3.

Solved Examples on Divisibility Rule of 3

**Example 1: Is the number 987654321 divisible by 3?

**Solution:

Sum of digits: 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45

Check if the sum is divisible by 3: 45 is divisible by 3.

Hence, 987654321 is divisible by 3.

**Example 2: Is the number 1001 divisible by 3?

**Solution:

Sum of the digits: 1 + 0 + 0 + 1 = 2

Check if the sum is divisible by 3: 2 is not divisible by 3.

Hence, 1001 is not divisible by 3.

**Example 3: Is the number 123456789 divisible by 3?

**Solution:

Sum of digits: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45

Check if the sum is divisible by 3: 45 is divisible by 3.

Hence, 123456789 is divisible by 3.

**Example 4: Is the number 780 divisible by 3?

**Solution:

Sum of the digits: 7 + 8 + 0 = 15

Check if the sum is divisible by 3: 15 is divisible by 3.

Hence, 780 is divisible by 3.

Divisibility Rule of 3 Worksheet

You can download this free worksheet on the divisibility rule of 3 with its answers here: Download Free Worksheet on Divisibility Rule of 3

**Also Check: **Practice Questions on Divisibility Rules