Divisibility rules — notes, question types & practice · Number System

01

Overview

A number $N$ is divisible by $d$ when $d$ divides $N$ exactly, leaving remainder $0$ . Divisibility tests let you check this without long division. This lesson covers the basic tests, composite divisors, missing digits, power rules, and reaching the nearest multiple.

02

The quick tests

Divisor	Test	Example
$2, 5, 10$	last digit even, $0$ or $5$ , $0$	$4,215$ by $5$
$4$	last two digits divisible by $4$	$7,316 \to 16$
$8$	last three digits divisible by $8$	$91,224 \to 224$
$3, 9$	digit sum divisible by $3$ or $9$	$6,453 \to 18$
$11$	odd-place sum minus even-place sum is $0$ or a multiple of $11$	see below
$7, 11, 13$	alternating sum of three-digit blocks	$276,276 \to 276 - 276 = 0$

Rule: For $11$ , number the digits from the right. Take (odd-place sum) $-$ (even-place sum). For $948475$ : odd places $5 + 4 + 4 = 13$ , even places $7 + 8 + 9 = 24$ , and $24 - 13 = 11$ , so it is divisible by $11$ .

03

Composite divisors split into co-prime parts

There is no direct rule of $72$ . Split the divisor into parts that share no factor, then check each part. Useful splits: $12 = 3 \times 4$ , $24 = 3 \times 8$ , $36 = 4 \times 9$ , $72 = 8 \times 9$ , $88 = 8 \times 11$ , $99 = 9 \times 11$ .

Watch: Never split $24$ as $4 \times 6$ . They share the factor $2$ : the number $36$ passes both tests yet leaves remainder $12$ with $24$ .

04

Missing digit questions

When digits are hidden as $x$ and $y$ , use the rule with the fewest choices first. The rule of $8$ pins the last three digits. The rule of $11$ gives one equation, the digit sum for $9$ gives another. Several pairs may work, but the asked quantity, usually $x + y$ , stays the same in every valid pair.

05

Divisibility of powers

Three rules cover almost every power question:

$a^n - b^n$ is divisible by $a - b$ for every $n$ .
$a^n - b^n$ is divisible by $a + b$ when $n$ is even.
$a^n + b^n$ is divisible by $a + b$ when $n$ is odd.

So $29^{18} - 15^{18}$ is divisible by both $29 - 15 = 14$ and $29 + 15 = 44$ , and $17^{25} + 23^{25}$ is divisible by $40$ because $25$ is odd.

For sums of powers of one base, pull out the smallest power: $3^{25} + 3^{26} + 3^{27} = 3^{25}(1 + 3 + 9) = 3^{25} \times 13$ .

06

Special digit patterns

A six-digit number written as a repeated three-digit block, like $276276$ , equals $\overline{abc} \times 1001$ , and $1001 = 7 \times 11 \times 13$ .
A three-digit number with equal digits, like $777 = 7 \times 111$ , is divisible by $37$ .
A two-digit number plus its reverse equals $11(a + b)$ ; the difference is $9(a - b)$ .
The product of three consecutive integers is divisible by $6$ .

07

Divisibility by seven

Cut the last digit, double it, and subtract from what remains. Repeat until the number is small. If the result is divisible by $7$ , so was the original: $3598 \to 359 - 16 = 343 = 7 \times 49$ .

08

Reaching the nearest multiple

To make $N$ divisible by $d$ , divide and find the remainder $r$ . Subtract $r$ to go down to the lower multiple, or add $d - r$ to go up. For $8357 \div 12$ the remainder is $5$ , so adding $7$ gives $8364 = 12 \times 697$ .

Example: Largest four-digit multiple of a divisor: take $9999$ , divide, and subtract the remainder from $9999$ .

09

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common3 practice Q

Missing digits for a composite divisor

How to spot it:

A number with blanks x and y is said to be divisible by 72, 88, 99 or 36, and the options are values of x + y or x - y.

72 = 8 \times 9,\quad 88 = 8 \times 11,\quad 99 = 9 \times 11

Method

Split the divisor into co-prime factors.
Apply the rule with fewer choices first: $8$ fixes the last three digits, $11$ gives one equation.
Feed that result into the second rule, the digit sum for $9$ or the alternating sum for $11$ .
List the working pairs; the asked sum or difference is the same in all of them.

Why it works:

A number divisible by two co-prime numbers is divisible by their product, and each rule cuts the possibilities.

Try this

If the five-digit number 6x8y4 is divisible by 72, find x + y.

Show solution

$72 = 8 \times 9$ . Rule of $8$ : the last three digits $8y4$ must divide by $8$ .
$824 = 8 \times 103$ and $864 = 8 \times 108$ , so $y = 2$ or $y = 6$ .
Rule of $9$ : digit sum $6 + x + 8 + y + 4 = 18 + x + y$ must divide by $9$ .
$y = 2$ forces $x = 7$ ; $y = 6$ forces $x = 3$ . Both give $x + y = 9$ .

Answer

9

Type 2very common2 practice Q

Divisibility by 11 with a missing digit

How to spot it:

A number like 7x2859 is divisible by 11 and one digit is hidden, or the options are numbers to pick from.

(\text{odd-place sum}) - (\text{even-place sum}) \in \{0, \pm 11, \pm 22, \ldots\}

Method

Number the digits from the right and add the odd places and the even places separately.
Subtract the two sums; the result must be $0$ or a multiple of $11$ .
Solve for the missing digit, which must stay between $0$ and $9$ .

Why it works:

Ten leaves remainder minus one with eleven, so place values flip between plus and minus.

Try this

If the number 7x2859 is divisible by 11, find x.

Show solution

From the right, odd places: $9 + 8 + x = 17 + x$ . Even places: $5 + 2 + 7 = 14$ .
Difference: $(17 + x) - 14 = 3 + x$ must be $0$ or $11$ .
$3 + x = 11$ gives $x = 8$ . Check: $782859 = 11 \times 71169$ .

Answer

8

Type 3very common3 practice Q

Divisibility of a power of the form a to the n minus b to the n

How to spot it:

Two big powers with the same exponent, added or subtracted, and the question asks which listed number divides them.

(a - b) \mid (a^n - b^n);\quad (a + b) \mid (a^n - b^n) \text{ for even } n;\quad (a + b) \mid (a^n + b^n) \text{ for odd } n

Method

Note whether the powers are added or subtracted, and whether $n$ is even or odd.
Compute $a - b$ and $a + b$ .
Match against the options: the difference always works; the sum needs the right parity of $n$ .

Why it works:

Factoring a to the n minus b to the n exposes both a minus b and, for even n, a plus b.

Try this

29 to the power 18 minus 15 to the power 18 is divisible by which of 14 and 44?

Show solution

$n = 18$ is even.
Divisible by $a - b = 14$ always, and by $a + b = 44$ because $n$ is even.
So both divide it.

Answer

Both 14 and 44

Type 4common2 practice Q

Sum of powers of the same base

How to spot it:

A sum like 3 to the 25 plus 3 to the 26 plus 3 to the 27 appears and the question asks what divides it.

a^m + a^{m+1} + a^{m+2} = a^m(1 + a + a^2)

Method

Take the smallest power common to every term outside a bracket.
Add the powers inside the bracket to get one small number.
The bracket value divides the whole sum.

Why it works:

Every term shares the smallest power, and the bracket becomes a plain number.

Try this

Find a number other than a power of 3 that divides 3 to the power 15 plus 3 to the power 16 plus 3 to the power 17.

Show solution

Factor: $3^{15}(1 + 3 + 9)$ .
$1 + 3 + 9 = 13$ .
So the sum equals $3^{15} \times 13$ and is divisible by $13$ .

Answer

13

Type 5very common3 practice Q

Least number to add or subtract to reach a multiple

How to spot it:

The question asks what least number must be added to or subtracted from N so the result is divisible by d, or for the largest n-digit multiple.

\text{add} = d - (N \bmod d), \quad \text{subtract} = N \bmod d

Method

Divide $N$ by $d$ and note the remainder $r$ .
To go down to the lower multiple, subtract $r$ .
To go up to the next multiple, add $d - r$ .
For the largest n-digit multiple, divide a string of nines and subtract its remainder.

Why it works:

Multiples of d sit d apart, and the remainder measures the distance to the lower one.

Try this

What least number must be added to 3286 to make it divisible by 29?

Show solution

$3286 = 29 \times 113 + 9$ , so the remainder is $9$ .
Add $29 - 9 = 20$ .
Check: $3306 = 29 \times 114$ .

Answer

20

Type 6common2 practice Q

Special forms: repeated blocks and equal digits

How to spot it:

A six-digit number repeats a three-digit block, or a three-digit number has equal digits, and the question asks what always divides it.

\overline{abcabc} = \overline{abc} \times 7 \times 11 \times 13, \qquad \overline{aaa} = a \times 3 \times 37

Method

Write the number as its block times a fixed multiplier.
Factor the multiplier: $1001 = 7 \times 11 \times 13$ and $111 = 3 \times 37$ .
Pick the option that divides the fixed multiplier.

Why it works:

The digit pattern builds a fixed multiplier no matter which digits are used.

Try this

Which of 7, 11 and 13 divide 276276?

Show solution

$276276 = 276 \times 1001$ .
$1001 = 7 \times 11 \times 13$ .
All three divide it.

Answer

All of 7, 11 and 13

Type 7occasional

Divisibility by seven

How to spot it:

The divisor 7 appears with a three- or four-digit number, often in an options question asking which numbers divide it.

N \to \left\lfloor \dfrac{N}{10} \right\rfloor - 2 \times (N \bmod 10)

Method

Cut the last digit and double it.
Subtract that from the remaining front part.
Repeat until the number has two or three digits.
If the result is divisible by $7$ , the original number is too.

Why it works:

Removing the last digit maps N to a smaller number with the same remainder apart from a factor, so divisibility survives.

Try this

Is 3598 divisible by 7?

Show solution

Cut $8$ , double it: $359 - 16 = 343$ .
$343 = 7 \times 49$ .
Yes, divisible by $7$ .

Answer

Yes, since 343 = 7 x 49

10

Formula sheet

Divisibility by 11

\left(\sum \text{odd-place digits}\right) - \left(\sum \text{even-place digits}\right) \in \{0, \pm 11, \pm 22, \ldots\}

Composite divisor

pq \mid N \iff p \mid N \text{ and } q \mid N, \quad \gcd(p, q) = 1

split into co-prime parts only

Difference of powers

(a - b) \mid (a^n - b^n) \text{ for all } n

Difference of even powers

(a + b) \mid (a^n - b^n) \text{ when } n \text{ is even}

Sum of odd powers

(a + b) \mid (a^n + b^n) \text{ when } n \text{ is odd}

Repeated block

\overline{abcabc} = \overline{abc} \times 1001 = \overline{abc} \times 7 \times 11 \times 13

Divisibility by seven

N \to \left\lfloor \dfrac{N}{10} \right\rfloor - 2 \times (N \bmod 10)

repeat until small

11

Shortcuts that save time

⚡ Split into co-prime factors

Break the divisor into parts with no common factor and test each part separately. Order the tests from the most restrictive first.

Example

If the five-digit number 37x84 is divisible by 12, find the smallest value of x.

Show solution

$12 = 3 \times 4$ . By $4$ : last two digits $84$ already pass, for any $x$ .
By $3$ : digit sum $3 + 7 + x + 8 + 4 = 22 + x$ must be a multiple of $3$ .
So $x \in \{2, 5, 8\}$ and the smallest is $2$ .

Answer

2

⚡ The repeated-block family

Any six-digit number abcabc equals abc times 1001, so it is always divisible by 7, 11 and 13.

Example

Which of 7, 11 and 13 divide 484484?

Show solution

$484484 = 484 \times 1001$ .
$1001 = 7 \times 11 \times 13$ .
All three divide it.

Answer

All of 7, 11 and 13

⚡ Sum of odd powers

For a to the n plus b to the n with odd n, check the options against a plus b first. No expansion is ever needed.

Example

Is 17 to the power 25 plus 23 to the power 25 divisible by 40?

Show solution

The exponent $25$ is odd, so $a^n + b^n$ is divisible by $a + b$ .
$17 + 23 = 40$ .
Yes, exactly divisible by $40$ .

Answer

Yes

12

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Splitting a composite divisor into parts that share a factor, like 12 = 2 x 6.

Use co-prime parts only: 12 = 3 x 4. A number can pass 2 and 6 yet fail 12.

Mistake 02

Checking only the last two digits for divisibility by 8.

The rule of 8 needs the last three digits.

Mistake 03

For 11, forgetting that the difference of the two sums can be 0 or negative.

Accept 0, 11, 22 and also -11 as valid differences.

Mistake 04

Using the (a + b) rule for a to the n plus b to the n when n is even.

That rule needs odd n. For even n only a minus b divides the difference.

Mistake 05

Multiplying the two rules' answers instead of checking both divisibilities.

For 72 = 8 x 9 the number must pass both tests; nothing is multiplied.

13

Quick revision

Read this the night before the exam.

Last digit tests $2, 5, 10$ ; last two digits for $4$ ; last three for $8$ .
Digit sum tests $3$ and $9$ ; alternating sums test $11$ .
Composite divisor: split into co-prime parts and test each part.
$(a-b)$ always divides $a^n - b^n$ ; $(a+b)$ divides it when $n$ is even; $(a+b)$ divides $a^n + b^n$ when $n$ is odd.
$\overline{abcabc} = \overline{abc} \times 7 \times 11 \times 13$ .
To reach a multiple: add $d - r$ or subtract $r$ .

14

Practice: 18 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 10 questions

Suggested time 9 min · wrong answers go to your mistake notebook automatically.