Divisibility rules are mathematical shortcuts that allow you to quickly determine if a number is divisible by another without performing actual division. These rules are based on patterns and properties of our decimal number system.
Fundamental Divisibility Rules:
Rule for 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8). This works because 10 ≡ 0 (mod 2), so only the units digit matters.
Rule for 3: A number is divisible by 3 if the sum of its digits is divisible by 3. This works because 10 ≡ 1 (mod 3), so each digit contributes its face value to the remainder.
Rule for 4: A number is divisible by 4 if its last two digits form a number divisible by 4. Since 100 ≡ 0 (mod 4), only the last two digits affect divisibility by 4.
Rule for 5: A number is divisible by 5 if it ends in 0 or 5. This is because 10 ≡ 0 (mod 5), making only the last digit relevant.
Rule for 6: A number is divisible by 6 if it's divisible by both 2 and 3, since 6 = 2 × 3 and gcd(2,3) = 1.
Advanced Rules:
Rule for 8: Check if the last three digits are divisible by 8, since 1000 ≡ 0 (mod 8).
Rule for 9: Sum all digits; if divisible by 9, so is the original number. This works because 10 ≡ 1 (mod 9).
Rule for 11: Alternating sum of digits. If this sum is divisible by 11, so is the number, because 10 ≡ -1 (mod 11).