Divisibility Rule of 12: Rules, Tricks and Examples

Divisibility Rule of 12 states that a number is divisible by 12 when it is divisible by both 3 and 4. This test combines the digit-sum rule for 3 with the last-two-digits rule for 4. The method helps check divisibility quickly without performing complete division.

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What Is the Divisibility Rule of 12?

A number is divisible by 12 if and only if it is divisible by both 3 and 4. Therefore, its digit sum must be divisible by 3, and the number formed by its last two digits must be divisible by 4.

Since 12 = 3 × 4 and 3 and 4 are relatively prime, both conditions are required. For example, 2,316 has digit sum 2 + 3 + 1 + 6 = 12, which is divisible by 3, and its last two digits, 16, are divisible by 4. Hence, 2,316 is divisible by 12.

Divisibility Rule of 12 Formula & Tricks

Important Formulas

Divisibility condition
N is divisible by 12 ⇔ N is divisible by 3 and N is divisible by 4

Apply both tests: the sum of the digits must be divisible by 3, and the last two digits must form a number divisible by 4.

Test for divisibility by 3
Sum of digits of N ≡ 0 (mod 3)

Add all the digits of the number. If the sum is 3, 6, 9, 12, or another multiple of 3, the number is divisible by 3.

Test for divisibility by 4
Last two digits of N ≡ 0 (mod 4)

Only the last two digits are considered. They must form a multiple of 4, including 00, 04, 08, 12, 16 and so on.

Quick Tricks

Use the two-part check

Check the digit sum first for divisibility by 3, then check the last two digits for divisibility by 4. The number passes the divisibility test for 12 only when both checks pass.

Example: For 1,452, the digit sum is 12, divisible by 3, and the last two digits are 52, divisible by 4. Therefore, 1,452 is divisible by 12.
Reject quickly when one test fails

If the digit sum is not divisible by 3 or the last two digits are not divisible by 4, the number cannot be divisible by 12.

Example: In 2,318, the digit sum is 14, which is not divisible by 3. Although 18 is not divisible by 4 as well, only one failed condition is enough to reject the number.

Divisibility Rule of 12 Concepts

Checking the Digit Sum for 3

The digit sum of a number must be a multiple of 3 for the number to be divisible by 12.

Add every digit of the number. If the sum is divisible by 3, the number passes the first part of the test. For 7,248, the digit sum is 7 + 2 + 4 + 8 = 21, so it is divisible by 3.

Example: For 5,634, the digit sum is 5 + 6 + 3 + 4 = 18. Since 18 is divisible by 3, 5,634 passes the divisibility-by-3 condition.

Checking the Last Two Digits for 4

The last two digits must form a number divisible by 4 for the number to be divisible by 12.

The divisibility of a number by 4 depends only on its last two digits. Thus, numbers ending in 00, 04, 08, 12, 16, 20, 24, 28 and other multiples of 4 pass this condition.

Example: In 8,316, the last two digits are 16, and 16 ÷ 4 = 4. Therefore, 8,316 passes the divisibility-by-4 condition.

Applying Both Conditions Together

A number passes the divisibility test for 12 only when both its digit sum is divisible by 3 and its last two digits are divisible by 4.

The two conditions must be checked independently. A number may satisfy one condition but fail the other, so checking only the digit sum or only the last two digits is insufficient.

Example: 2,124 has digit sum 2 + 1 + 2 + 4 = 9, divisible by 3, and its last two digits are 24, divisible by 4. Hence, 2,124 is divisible by 12 because 2,124 ÷ 12 = 177.

Numbers Ending in Zero

A number ending in 00 is divisible by 4, but it is divisible by 12 only if its digit sum is also divisible by 3.

The last two digits 00 satisfy the divisibility-by-4 condition automatically. The digit-sum test still decides whether the number is divisible by 3.

Example: 3,600 has digit sum 3 + 6 = 9 and ends in 00. Both conditions hold, so 3,600 is divisible by 12.

Divisibility Rule of 12 Video Lessons

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14 Lessons
Lesson 1 of 14 Quick Revision

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Practice Divisibility Rule of 12 Questions

Practise published questions related to this topic.

1√(1.44 × 0.0225) = ?→ 2Three numbers are in the ratio 4 : 3 : 13, and their LCM is 4524. Their HCF is:→ 3√0.09/0.6 × 0.4/√0.36 × 1.8/√0.36 = ?→ 4Least Common Multiple of 11 and 30 is X, Highest Common Factor of 11 and 30 is Y. Then what is the value of (X - 2Y)?→ 5Rama made necklaces with either 24 beads, 36 beads or 40 beads, such that no bead was left over. What is the least number of beads Rama could have?→ 6Find the Highest Common Factor (HCF) of 130, 140, and 185 .→ 7The H.C.F. and L.C.M. of two numbers are 25 and 500, respectively. If the first number is divided by 2, the quotient is 50 and the remainder is zero. Find the second number.→ 8The sum of a number and its positive square root is 30. What is the value of the number?→ 9The number 657792025 is divisible by which of the following numbers?→ 10A teacher has 84 chocolates and 108 candies. She wants to distribute them equally among the maximum number of students so that each student gets the same number of chocolates and the same number of candies. Find the maximum number of students and how many chocolates and candies each will get.→

Divisibility Rule of 12 Quick Quiz

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Quick Revision Notes

Divisibility Rule of 12: Quick Revision

Use these checks to test any whole number for divisibility by 12.

  • 12 = 3 × 4, and 3 and 4 are relatively prime.
  • A number is divisible by 12 if it is divisible by both 3 and 4.
  • For divisibility by 3, the sum of all digits must be divisible by 3.
  • For divisibility by 4, the last two digits must form a multiple of 4.
  • A number ending in 00 automatically passes the test for 4, but its digit sum must still be divisible by 3.
  • If either condition fails, the number is not divisible by 12.

Divisibility Rule of 12 FAQs

What is the divisibility rule of 12?

A number is divisible by 12 when it is divisible by both 3 and 4. Its digit sum must be divisible by 3, and its last two digits must be divisible by 4.

Is 1,728 divisible by 12?

Yes. Its digit sum is 1 + 7 + 2 + 8 = 18, divisible by 3, and its last two digits, 28, are divisible by 4. Therefore, 1,728 is divisible by 12.

Is a number divisible by 12 if its digit sum is divisible by 3?

Not necessarily. The last two digits must also be divisible by 4. For example, 1,230 has digit sum 6 but its last two digits, 30, are not divisible by 4, so it is not divisible by 12.

Is a number divisible by 12 if its last two digits are divisible by 4?

Not necessarily. Its digit sum must also be divisible by 3. For example, 1,124 ends in 24 but has digit sum 8, so it is not divisible by 12.

How can divisibility by 12 be checked for a number ending in 00?

The number automatically satisfies the divisibility-by-4 condition. Check whether its digit sum is divisible by 3; if yes, the number is divisible by 12.

What is the smallest positive number divisible by both 3 and 4?

The smallest positive number is 12. Since 12 is the least common multiple of 3 and 4, every common multiple of 3 and 4 is a multiple of 12.

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