Divisibility Rules: Practice Questions & Notes

Divisibility is the ability of one number to divide another without leaving a remainder. It's the bedrock of number theory, helping in shortcuts, simplifications, and logic-based MCQs.

Key Idea

A number $A$ is divisible by $B$ if:

A \mod B = 0 \quad \text{or} \quad \frac{A}{B} \text{ is a whole number}

This topic focuses on divisibility rules, applications, and common traps.

Divisibility Rules (1 to 20)

Divisor	Divisibility Rule
1	Every number is divisible by 1
2	Last digit is even (0, 2, 4, 6, 8)
3	Sum of digits divisible by 3
4	Last 2 digits divisible by 4
5	Ends in 0 or 5
6	Divisible by both 2 and 3
7	Double the last digit, subtract from the rest; result divisible by 7
8	Last 3 digits divisible by 8
9	Sum of digits divisible by 9
10	Ends in 0
11	Alternating sum of digits divisible by 11
12	Divisible by both 3 and 4
13	Remove the last digit, multiply it by 9, subtract from the rest; check result
14	Divisible by both 2 and 7
15	Divisible by both 3 and 5
16	Last 4 digits divisible by 16
17	Subtract 5 × last digit from the rest; result divisible by 17
18	Divisible by both 2 and 9
19	Double the last digit, add to the rest; result divisible by 19
20	Ends in 0, and second last digit must be even

Conceptual Shortcuts

Divisibility via Digital Sum

Quick test for divisibility by 3 and 9:

$123456$ : Sum = 1+2+3+4+5+6 = 21 → divisible by 3, not 9

Using Prime Factorization

To check divisibility by a composite number, factor it and apply rules:

Is a number divisible by 36?
36 = 4 × 9 → Check divisibility by both

Special Patterns

Divisibility by 7, 13, 17, 19, etc.

Most exams test these using smart logic or modular arithmetic:

Divisibility by 7 (Short-Cut Rule)

Example: Is 672 divisible by 7?

Double last digit → 2 × 2 = 4
Subtract from rest: 67 − 4 = 63
63 is divisible by 7 → So, 672 is divisible

Divisibility by 11 (Alternating Sum Rule)

If $a_1a_2a_3a_4...$ , then compute:

(a_1 - a_2 + a_3 - a_4 + \dots)

If the result is divisible by 11, so is the number.

Example: 121
$1 - 2 + 1 = 0 \Rightarrow$ Divisible

Common Misconceptions

Mistake	Correction
Assuming divisibility by 6 means divisible by 3 or 2	Must be divisible by both 2 and 3
Only using last digit to check for 3 or 9	Use sum of digits
Divisibility by 11 means alternating digits must be equal	No — use the alternating sum rule

Applications in Exams

Option elimination
Finding remainders
Number property questions
Puzzles and reasoning

Useful Tricks

Fast Checking for Large Numbers

Check last few digits only when rules require.

Divisible by 4 → Check last 2 digits
Divisible by 8 → Check last 3 digits
Divisible by 16 → Check last 4 digits

Cyclic Digital Sum Technique

Used to reduce big numbers:

987654321 \rightarrow \text{Digital sum} = 45 \rightarrow 4 + 5 = 9 \Rightarrow \text{Divisible by 9}

Practice Problems

Q1. Is 142857 divisible by 3?

Sum of digits = 27 → Divisible by 3 → Yes

Q2. Is 123456 divisible by 8?

Last 3 digits = 456
456 ÷ 8 = 57 → Exact division → Yes

Q3. Find smallest 3-digit number divisible by 7

First 3-digit number = 100
100 ÷ 7 ≈ 14.28 → Next multiple = $15 \times 7 = 105$

Q4. Which of these is divisible by 11: 3146, 5285, 1331?

3146 → 3−1+4−6 = 0 → Yes
5285 → 5−2+8−5 = 6 → No
1331 → 1−3+3−1 = 0 → Yes

Visual Aid

Check last digits → Shortcuts (2, 5, 10, 4, 8)
Sum of digits → Shortcuts (3, 9)
Alternating sum → Shortcut (11)
Special pattern rules → 7, 13, 17, 19