Check if one number divides evenly into another. If the remainder is 0, the number is divisible. For example, 12 ÷ 3 = 4 with remainder 0, so 12 is divisible by 3.
The divisibility calculator tests whether one number divides evenly into another. Choose your check mode: Single Divisor (test one specific divisor) or Common Divisors (test multiple divisors 2, 3, 4, 5, 6, 8, 9, 10, 12 at once).
In Single Divisor mode, enter the number to test in "Number to Check" and the divisor in "Divide By". For example, enter 24 and 6 to verify that 24 is divisible by 6 (result: 24÷6=4, remainder 0).
Try testing 25÷6: enter 25 and 6. The calculator shows "No, 25 is not divisible by 6. Remainder: 1" because 25÷6=4 with remainder 1. Any non-zero remainder means not divisible.
Click Check to see results. For Single Divisor, you get a yes/no answer with the quotient or remainder. For Common Divisors, you see a color-coded grid: green (✓) for divisible, red (✗) for not divisible. Use Reset to clear and test new numbers.
Using Single Divisor Mode
Select Single Divisor to test if one specific number divides evenly into another. This mode answers questions like "Is 48 divisible by 7?" or "Does 5 divide evenly into 100?"
Enter your number in "Number to Check" and the potential divisor in "Divide By". The calculator performs division and checks if the remainder is zero. For 48÷6: quotient = 8, remainder = 0, so yes, 48 is divisible by 6.
The result shows three pieces of information: divisibility status (yes/no), the quotient if divisible, or the remainder if not. Try 50÷7: "No, not divisible. Remainder: 1" because 50=7×7+1.
This mode is useful for verifying calculations, checking if numbers are multiples, or testing divisibility before simplifying fractions. For 4836, you might test if both are divisible by 6, 8, or 12 to find the greatest common factor.
Using Common Divisors Mode
Switch to Common Divisors to test divisibility by 2,3,4,5,6,8,9,10, and 12 simultaneously. This mode reveals patterns and properties of your number at a glance. Enter just one number and click Check.
The result displays a grid with nine boxes. Green boxes with ✓ show divisors that work; red boxes with ✗ show divisors that don't. Try entering 36: you'll see green for 2,3,4,6,9,12 and red for 5,8,10.
This visual grid helps identify number properties quickly. If 2,4,8 are all green, your number is divisible by all powers of 2 up to 8. If only 2 is green but 4 is red, your number is 2×odd.
Common Divisors mode is excellent for learning divisibility rules, finding quick factorizations, or checking if a number is composite (if multiple divisors show green, it's not prime). Try 120 to see it's divisible by all nine common divisors.
Understanding Divisibility
A number a is divisible by b if a÷b gives a whole number with remainder 0. In mathematical notation: a÷b=q with a=b×q for some integer q. For example, 24 is divisible by 6 because 24=6×4.
Divisibility is the foundation of many number theory concepts. If a is divisible by b, then b is a factor of a, and a is a multiple of b. The number 12 is divisible by 1,2,3,4,6,12—these are all factors of 12.
Testing divisibility is equivalent to checking if the remainder (modulo) is zero. The calculator uses the modulo operation: if amodb=0, then a is divisible by b. This is why 15mod5=0 confirms 15 is divisible by 5.
Understanding divisibility helps with fraction simplification, finding common denominators, prime factorization, and solving modular arithmetic problems. It's a fundamental skill in elementary number theory and practical mathematics.
Divisibility Rules and Patterns
Divisibility rules provide shortcuts for testing divisibility without full division. Divisible by 2: last digit is even (0,2,4,6,8). Try 48: last digit 8 is even, so divisible by 2.
Divisible by 3: sum of digits is divisible by 3. For 123: 1+2+3=6, and 6 is divisible by 3, so 123 is divisible by 3. Divisible by 5: last digit is 0 or 5. Test 125: ends in 5, so divisible by 5.
Divisible by 4: last two digits form a number divisible by 4. For 316: the last two digits are 16, and 16÷4=4, so 316 is divisible by 4. Divisible by 9: sum of digits is divisible by 9. Try 729: 7+2+9=18, divisible by 9.
Divisible by 10: last digit is 0. Divisible by 6: divisible by both 2 and 3. Divisible by 12: divisible by both 3 and 4. These rules come from properties of base-10 number system and modular arithmetic.
Reading Results and Interpreting Output
In Single Divisor mode, results appear as colored text. Green text starting with "Yes" means divisible, showing the quotient: "Yes, 24 is divisible by 6. Result: 4". Orange/yellow "No" means not divisible, showing the remainder.
The remainder tells you how much is left over after division. For "27 is not divisible by 5. Remainder: 2", this means 27=5×5+2. The remainder is always less than the divisor.
In Common Divisors mode, the grid uses color coding. Green boxes with ✓ indicate divisibility; red boxes with ✗ indicate non-divisibility. Each box is labeled with its divisor number (2, 3, 4, etc.) for easy reading.
Error messages appear in red. "Error: Enter a valid number" means you left a field blank or entered non-numeric text. "Error: Enter a valid divisor (non-zero integer)" means the divisor is 0, blank, or not an integer. Division by zero is undefined.
Applications in Fraction Simplification
Divisibility testing is essential for reducing fractions. To simplify 3624, test which numbers divide both 24 and 36. Using Common Divisors mode: both are divisible by 2,3,4,6,12.
The greatest common divisor is 12, so divide both: 36÷1224÷12=32. The divisibility calculator helps you quickly identify potential common divisors without trial-and-error division.
For complex fractions like 192144, test both numbers with Common Divisors mode. You'll find both are divisible by 2,3,4,6,8,12. Testing these as potential GCF candidates, you discover GCF = 48, giving 192144=43.
Divisibility also helps verify if a fraction is already in lowest terms. If numerator and denominator share no common divisors (other than 1), the fraction is fully reduced. Test 2815: they share no common divisors, so it's already simplified.
Using Divisibility for Prime Testing
Divisibility testing helps identify composite numbers (non-prime). If a number is divisible by any number other than 1 and itself, it's composite. Use Common Divisors mode to quickly check.
Enter 77 in Common Divisors mode: no common divisors show green, but this doesn't prove it's prime—you'd need to test all numbers up to 77≈8.77. However, 77=7×11 is actually composite (divisible by 7 and 11).
For small numbers, Common Divisors mode catches most composites. If any divisor shows green (except cases like 2 dividing even numbers you already know about), the number is definitely composite. Try 91: no common divisors, but 91=7×13.
To conclusively test primality, you'd need a prime number checker that tests all divisors up to the square root. Divisibility testing is a first step: if divisible by 2,3,5, etc., definitely not prime. If not, further testing needed.
Divisibility and Modular Arithmetic
Divisibility is the foundation of modular arithmetic (mod or remainder arithmetic). Saying "a is divisible by b" is equivalent to "a≡0(modb)" (a is congruent to 0 modulo b).
The remainder from divisibility testing is the modulo value. When the calculator shows "Remainder: 3" for 23÷5, this means 23mod5=3 or 23≡3(mod5). Modular arithmetic uses these remainders for clock arithmetic, cryptography, and number theory.
Divisibility rules are modular arithmetic shortcuts. "Last digit even" for divisibility by 2 works because 10≡0(mod2), so only the last digit matters. Similarly, "sum of digits" for divisibility by 3 works because 10≡1(mod3).
Understanding the connection between divisibility and modulo helps solve congruence problems, Chinese Remainder Theorem applications, and cryptographic algorithms like RSA. The divisibility calculator shows modulo results (remainders) when numbers aren't divisible.
Related Calculators and Concepts
Modulo Calculator - Calculate remainders directly. Complements divisibility testing by showing amodb for any numbers. Divisibility occurs when modulo equals zero.
Factoring Calculator - Find all factors of a number. Every factor represents a number that divides evenly. Combines divisibility testing with complete factor enumeration.
GCF Calculator - Find greatest common factor. Uses divisibility to identify which numbers divide into multiple inputs. Essential for fraction simplification.
Prime Number Checker - Test if a number is prime. A number is prime only if it's divisible by exactly two numbers: 1 and itself.
LCM Calculator - Find least common multiple. The LCM is divisible by all input numbers—it's the smallest such number. Divisibility is central to understanding multiples.