Sum-of-Digits Tests
The sum-of-digits divisibility tests quickly determine whether large numbers are divisible by 3 or 9. In Grade 6 Saxon Math Course 1, students add all the digits of a number: if the digit sum is divisible by 3, so is the number; if divisible by 9, so is the number. For 4,761: digit sum = 4 + 7 + 6 + 1 = 18; 18 is divisible by both 9 and 3, so 4,761 is divisible by both. This shortcut is used in prime factorization and GCF/LCM calculations to identify factors quickly.
Key Concepts
Property Add the digits of the number and inspect the total. A number is divisible by: $3$ if the sum of the digits is divisible by $3$. $9$ if the sum of the digits is divisible by $9$.
Examples Is $762$ divisible by $3$? Yes, because $7+6+2=15$, and $15$ is divisible by $3$. Is $2745$ divisible by $9$? Yes, because $2+7+4+5=18$, and $18$ is divisible by $9$. Is $128$ divisible by $3$? No, because $1+2+8=11$, and $11$ is not divisible by $3$.
Explanation This test is like a magic trick! To see if a big number is divisible by $3$ or $9$, just add up all its individual digits. If that little sum is divisible by $3$ or $9$, then the original huge number is too! It is a simple way to handle large numbers without breaking a sweat. Teamwork makes the dream work!
Common Questions
What are the sum-of-digits divisibility tests?
Add all digits of the number. If the sum is divisible by 3, the number is divisible by 3. If the sum is divisible by 9, the number is divisible by 9.
Is 4,761 divisible by 9?
Digit sum: 4 + 7 + 6 + 1 = 18. 18 ÷ 9 = 2. Yes, 4,761 is divisible by 9.
Is 5,234 divisible by 3?
Digit sum: 5 + 2 + 3 + 4 = 14. 14 is not divisible by 3, so 5,234 is not divisible by 3.
If a number is divisible by 9, is it also divisible by 3?
Yes. Every multiple of 9 is also a multiple of 3, because 9 = 3 × 3.
Why do the sum-of-digits tests work?
In base 10, each digit position is congruent to 1 (mod 9), so the digit sum has the same remainder when divided by 9 as the original number does.