An integer is divisible by 3 if the sum of its digits is divisible by 3. This simple rule allows for quick determination of divisibility without performing the actual division.
Understanding the Divisibility Rule of 3
The divisibility rule for 3 stems from the properties of modular arithmetic. Sources like Cuemath and Byju's clearly explain this rule. Essentially, any number can be expressed as a sum of its digits multiplied by powers of 10. Since 10 leaves a remainder of 1 when divided by 3 (10 = 3*3 + 1), the remainder when a number is divided by 3 is the same as the remainder when the sum of its digits is divided by 3.
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Example 1: Consider the number 123. The sum of its digits is 1 + 2 + 3 = 6. Since 6 is divisible by 3, 123 is also divisible by 3.
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Example 2: Consider the number 475. The sum of its digits is 4 + 7 + 5 = 16. Since 16 is not divisible by 3, 475 is not divisible by 3.
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Example 3 (Large Number): As mentioned in the YouTube video snippet, even large numbers can be checked easily using this method. Add the digits, and if that sum is divisible by 3, the original number is divisible by 3.
Programmatic Approach
Many programming languages offer the modulo operator (%), which provides the remainder after division. As shown in a Stack Overflow example (https://stackoverflow.com/questions/6647296/how-to-check-if-an-integer-can-be-divided-by-3), you can check divisibility by 3 using this:
if (i % 3 == 0) {
// The number i is divisible by 3
}However, for very large numbers, summing the digits and checking the sum's divisibility is significantly more efficient than direct modulo operation.
Conclusion
The divisibility rule offers a quick and easy method to determine if a number is divisible by 3. No complex calculations are needed; simply add the digits and check the result for divisibility by 3.
