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Divisibility Table

💡 How to Use This Tool▼

Select Divisors

Click any ÷ button to highlight all numbers divisible by that value. Select multiple divisors to see numbers divisible by any of them. Click again to deselect.

Explore Numbers

Hover over any number in the grid to see all its divisors (1-12) and the divisibility rule explanation for each one.

Learn the Rules

The tooltip shows why each number is divisible—digit sums for 3 and 9, last digits for 2, 5, and 10, and combined rules for 6 and 12.

Select divisors to highlight • Hover any number for details

Click to select divisors • Click again to deselect • Mix and match!

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Selecting Divisors
Hovering for Divisibility Details
Understanding the Grid Layout
Divisibility Rule for 2
Divisibility Rule for 3
Divisibility Rule for 4
Divisibility Rule for 5
Divisibility Rule for 6
Divisibility Rule for 9
Divisibility Rule for 10
Divisibility Rule for 11
Divisibility Rule for 12
What is Divisibility?
Why Divisibility Rules Work
Related Concepts






Selecting Divisors

Click any divisor button (÷1 through ÷12) at the top of the grid to highlight all numbers from 0 to 100 that are divisible by that value. The button turns blue when active, and matching numbers in the grid become highlighted with a light blue background.

You can select multiple divisors simultaneously. When multiple divisors are selected, the grid highlights numbers divisible by any of the selected values. This lets you compare divisibility patterns—for example, selecting both ÷2 and ÷3 shows all even numbers plus all multiples of 3.

Click a selected divisor again to deselect it. Use the red ✕ button to clear all selections at once and reset the grid to its default state.

Hovering for Divisibility Details

Hover over any number in the grid to see a detailed tooltip. The tooltip displays all divisors from 1 to 12 that evenly divide that number, along with an explanation of why each divisibility rule works.

For example, hovering over 36 shows it's divisible by 1, 2, 3, 4, 6, 9, and 12. Each divisor includes a brief explanation: "Last digit 6 is even" for ÷2, or "3+6=9, ÷9✓" for ÷9.

The hovered cell enlarges and turns blue, making it easy to track which number you're examining. The tooltip appears above the number (or below for 0) and follows your cursor as you explore different cells.

Understanding the Grid Layout

The grid displays numbers 0 through 100 arranged in 17 columns. This layout makes it easy to spot vertical patterns when divisors are selected.

Number 0 is special: it's divisible by every number (since 0 ÷ n = 0 for any n). The tooltip for 0 shows all twelve divisors with the explanation "0 ÷ anything = 0."

Number 1 is only divisible by itself. All other numbers have at least two divisors (1 and the number itself), with composite numbers having additional divisors between them.

Divisibility Rule for 2

A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8). This is the simplest divisibility test because you only need to check one digit.

When you select ÷2, exactly half the grid lights up—all the even numbers. Notice how they form a checkerboard-like pattern in the grid.

The tooltip shows "Last digit X is even" for any number divisible by 2. For example, 48 shows "Last digit 8 is even" to explain why 48 ÷ 2 = 24 works.

Divisibility Rule for 3

A number is divisible by 3 if the sum of its digits is divisible by 3. This rule works because 10 ≡ 1 (mod 3), so each digit contributes its face value to the remainder.

The tooltip shows the digit sum calculation. For 87, it displays "8+7=15, ÷3✓" because 15 is divisible by 3 (15 = 3 × 5), confirming that 87 ÷ 3 = 29.

Try selecting ÷3 and notice how multiples of 3 are spread throughout the grid—every third number starting from 0, 3, 6, 9, and so on.

Divisibility Rule for 4

A number is divisible by 4 if its last two digits form a number divisible by 4. This works because 100 is divisible by 4, so only the final two digits matter.

For two-digit numbers in this grid, you check the entire number. The tooltip shows "XX ÷ 4 = Y" with the exact division. For example, 76 shows "76 ÷ 4 = 19."

Select ÷4 to see that every fourth number is highlighted. Compare with ÷2—all numbers divisible by 4 are also divisible by 2, but not vice versa.

Divisibility Rule for 5

A number is divisible by 5 if its last digit is 0 or 5. Like the rule for 2, you only need to check the final digit.

The tooltip displays "Ends in X" where X is 0 or 5. For 45, it shows "Ends in 5" to confirm divisibility.

When you select ÷5, you'll see two columns light up in the grid—numbers ending in 0 and numbers ending in 5. This creates a distinctive striped pattern that's easy to recognize.

Divisibility Rule for 6

A number is divisible by 6 if it's divisible by both 2 and 3. Since 6 = 2 × 3 and these factors share no common divisors, both conditions must be met.

The tooltip shows "÷2 and ÷3 both work" for numbers divisible by 6. This reminds you that 6 is a composite rule requiring two separate checks.

Try selecting ÷6, then compare by selecting ÷2 and ÷3 together. The ÷6 highlights are exactly where the ÷2 and ÷3 patterns overlap—numbers that satisfy both conditions.

Divisibility Rule for 9

A number is divisible by 9 if the sum of its digits is divisible by 9. This is the same principle as the rule for 3, but with a stricter requirement.

The tooltip shows the digit sum, just like for 3. For 81, it displays "8+1=9, ÷9✓" because 9 is divisible by 9.

Notice that all numbers divisible by 9 are also divisible by 3 (since 9 = 3 × 3), but not all multiples of 3 are multiples of 9. Select both ÷3 and ÷9 to see how ÷9 highlights form a subset of ÷3 highlights.

Divisibility Rule for 10

A number is divisible by 10 if its last digit is 0. This is the strictest single-digit test—only numbers ending in zero qualify.

The tooltip simply shows "Ends in 0" for these numbers. In our 0-100 grid, exactly eleven numbers are divisible by 10: 0, 10, 20, 30, 40, 50, 60, 70, 80, 90, and 100.

Since 10 = 2 × 5, every number divisible by 10 is also divisible by both 2 and 5. Select all three to verify this relationship.

Divisibility Rule for 11

A number is divisible by 11 if the alternating sum of its digits is divisible by 11 (including 0). Starting from the leftmost digit, alternate between adding and subtracting.

For 99, the alternating sum is 9 - 9 = 0, which is divisible by 11. The tooltip shows "Alt sum = 0, ÷11✓."

For 121 (outside our grid), it would be 1 - 2 + 1 = 0. This rule is trickier than others, making the tooltip explanations especially helpful for understanding why certain numbers qualify.

Divisibility Rule for 12

A number is divisible by 12 if it's divisible by both 3 and 4. Since 12 = 3 × 4 and these factors share no common prime factors, both conditions must hold.

The tooltip shows "÷3 and ÷4 both work" for numbers divisible by 12. Like the rule for 6, this is a composite test.

In the 0-100 range, multiples of 12 are: 0, 12, 24, 36, 48, 60, 72, 84, and 96. Select ÷12 to highlight them, then verify by selecting ÷3 and ÷4 together—the intersection matches exactly.

What is Divisibility?

Divisibility means one integer divides another with no remainder. We say "a is divisible by b" (written a | b or b divides a) when a ÷ b produces a whole number.

For example, 24 is divisible by 6 because 24 ÷ 6 = 4 exactly. But 25 is not divisible by 6 because 25 ÷ 6 = 4 remainder 1.

Divisibility rules provide shortcuts to determine divisibility without performing full division. Instead of computing 738 ÷ 9, you can add digits: 7 + 3 + 8 = 18, and since 18 is divisible by 9, so is 738.

Why Divisibility Rules Work

Divisibility rules exploit patterns in our base-10 number system. Every number can be written as a sum of its digits times powers of 10:

247=2×100+4×10+7×1247 = 2 \times 100 + 4 \times 10 + 7 \times 1


For divisibility by 9, note that 10 ≡ 1 (mod 9), 100 ≡ 1 (mod 9), and so on. This means each power of 10 contributes just its digit's value to the remainder when divided by 9. So 247 mod 9 equals (2 + 4 + 7) mod 9 = 13 mod 9 = 4.

Similar reasoning explains other rules. For divisibility by 4, since 100 ≡ 0 (mod 4), only the last two digits matter.

Related Concepts

Divisibility connects to many fundamental arithmetic and number theory concepts:

• Factors and Multiples - If a is divisible by b, then b is a factor of a, and a is a multiple of b

• Prime Numbers - Numbers divisible only by 1 and themselves. Primes have exactly two divisors

• Greatest Common Divisor (GCD) - The largest number that divides two integers. Found using divisibility relationships

• Least Common Multiple (LCM) - The smallest number divisible by two integers

• Prime Factorization - Breaking numbers into prime factors reveals all divisibility relationships

• Modular Arithmetic - Divisibility rules are applications of modular arithmetic, where a is divisible by b means a ≡ 0 (mod b)