Divisibility Quick Kit: quick revision, 14 bites

Why Divisibility Rules Matter

Quick checks without long division.

**Divisibility rules** are shortcuts to determine whether one integer divides another without performing full division. They are essential for simplifying fractions, finding factors, and solving problems in number theory.

Takeaway. Divisibility rules turn tedious division into quick mental checks.

What Does 'Divisible' Mean?

Exact division with no remainder.

An integer $a$ is **divisible** by an integer $b$ (where $b \neq 0$) if there exists an integer $k$ such that $a = b \times k$. In other words, dividing $a$ by $b$ leaves a remainder of $0$.

Takeaway. Divisibility means the remainder is zero.

Divisibility by 2, 5, and 10

Check the last digit.

Takeaway. For 2, 5, and 10, just look at the last digit.

Divisibility by 3 and 9

Sum of digits tells the story.

A number is divisible by **3** if the sum of its digits is divisible by 3. Similarly, divisible by **9** if the sum of digits is divisible by 9.

Takeaway. Sum of digits rule works for 3 and 9.

Divisibility by 4 and 8

Look at the last two or three digits.

Takeaway. For 4 and 8, check the last two or three digits respectively.

Divisibility by 6

Must pass both 2 and 3.

A number is divisible by **6** if it is divisible by **both 2 and 3**. This is because $6 = 2 \times 3$ and 2 and 3 are coprime.

Takeaway. Divisible by 6 means divisible by both 2 and 3.

Divisibility by 11

Alternating sum of digits.

A number is divisible by **11** if the difference between the sum of digits in odd positions and the sum of digits in even positions is a multiple of 11 (including 0).

Takeaway. Use alternating sum for divisibility by 11.

Divisibility by 7

Double the last digit and subtract.

To check divisibility by **7**, double the last digit and subtract it from the rest of the number. Repeat until you get a small number that you know is or isn't divisible by 7.

Takeaway. The double-and-subtract method works for 7.

Divisibility by 12

Combine rules for 3 and 4.

A number is divisible by **12** if it is divisible by both **3 and 4**. Since 3 and 4 are coprime, this condition is necessary and sufficient.

Takeaway. Divisible by 12 means divisible by both 3 and 4.

Don't Confuse 6 and 12 Rules

6 needs 2&3; 12 needs 3&4.

A common error is thinking a number divisible by 6 is also divisible by 12, or that 12 requires divisibility by 2 and 6. Remember: **12 = 3 × 4**, not 2 × 6 (since 2 and 6 are not coprime).

Takeaway. For composite divisors, use coprime factor pairs.

Using Divisibility in Prime Factorization

Break numbers down efficiently.

Divisibility rules help quickly test small prime factors, making **prime factorization** faster. For example, to factor 252, check divisibility by 2 (yes), 3 (sum=9, yes), etc., to get $252 = 2^2 \times 3^2 \times 7$.

Takeaway. Divisibility rules speed up prime factorization.

Real-World Use: Splitting Bills

Fair sharing without a calculator.

When splitting a bill among friends, divisibility rules help quickly check if an amount can be divided evenly. For instance, if the bill is $123, you can see it's divisible by 3 (sum=6) but not by 2, so splitting among 3 works but not 2.

Takeaway. Divisibility rules aid quick mental math in daily life.

NCERT Exam Tip

Memorize rules for 2,3,5,9,10,11.

In NCERT exams, questions often ask to check divisibility by small numbers. **Memorize** the rules for 2, 3, 4, 5, 6, 8, 9, 10, and 11. Practice with examples from your textbook.

Takeaway. Focus on the most common divisibility rules for exams.

Memory Trick: Last Digit vs Sum

Ends vs adds.

Remember: **Ends** for 2,5,10 (look at last digit), **Adds** for 3 and 9 (sum of digits). This mnemonic helps recall the two main categories.

Takeaway. Ends for 2,5,10; adds for 3,9.