AFCAT Numerical Ability · Number System

Divisibility Rules from 2 to 12, and Counting Multiples

Decide whether a number divides exactly without dividing: read its last digits, its digit sum or its alternating digit sum, and combine rules for 6, 12, 15 and the like.

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Why this matters

Most questions here name a number and a divisor and ask yes or no, so each rule has to be exact. The HARD ones count or add the multiples in a range, or ask what to add to make a number divisible, and that is where a missed 1 in a count or a remainder read the wrong way costs the mark.

Concept 1 of 6: Rules that read the last digits: 2, 4, 5, 8, 10 and 25

10, 100 and 1000 are already divisible by these numbers, so everything to the left of the last one, two or three digits divides exactly. Only the tail can leave a remainder.

Definition

Look at as many digits from the right as the rule says, and test that small number. For 2, 5 and 10 one digit is enough, for 4 and 25 two, for 8 three.

Divisor	Test	Example that passes
2	Last digit is 0, 2, 4, 6 or 8	5,318
4	Last two digits form a multiple of 4	7,316 (16) An even last digit is not enough: 1,326 ends in 26, which is not a multiple of 4.
8	Last three digits form a multiple of 8	41,128 (128)
5	Last digit is 0 or 5	6,035
10	Last digit is 0	4,560
25	Last two digits are 00, 25, 50 or 75	3,175

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Example 1 · Number System · Divisibility Rules

Which number is NOT divisible by

4

Two digits are not enough for 8

3,140 ends in 40, a multiple of 8, but the rule for 8 reads three digits: 140 is not a multiple of 8, so 3,140 is not divisible by 8.

Drill 6 more on rules that read the last digits: 2, 4, 5, 8, 10 and 25

Concept 2 of 6: Rules that add the digits: 3 and 9

Every power of 10 is one more than a multiple of 9 (

10 = 9 + 1

100 = 99 + 1

). So a number leaves the same remainder on division by 9, or by 3, as the sum of its digits does.

Definition

3: the digit sum is a multiple of 3.
9: the digit sum is a multiple of 9.
If the digit sum is large, add its digits again.
Divisible by 9 means divisible by 3, but not the other way round: 2,145 has digit sum 12, so it is divisible by 3 and not by 9.

Digit-sum rule

N \text{ divisible by } 9 \;\text{means}\; \text{its digit sum is divisible by } 9

Worked example

Is 87,354 divisible by 3? By 9?

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Practice questionEasy

Example 2 · Number System · Divisibility Rules

Which number is NOT divisible by

3

Divisible by 3 is not divisible by 9

99,999,996 has digit sum 69, a multiple of 3 but not of 9. It is divisible by 3 only.

Drill 3 more on rules that add the digits: 3 and 9

Concept 3 of 6: Rules for 7 and 11

For 7, cut off the last digit, double it and subtract it from what is left; the result is divisible by 7 exactly when the number was. For 11, add and subtract the digits in turn from the right, because

10

is one less than 11.

Definition

7: remove the last digit, subtract twice it from the rest, and repeat until the number is small.
11: starting from the units digit, add, subtract, add, ... the digits. The number is divisible by 11 when this alternating sum is 0 or a multiple of 11 (11, 22, $-11$ , ...).
The alternating sum also gives the remainder on division by 11 when it is not a multiple: a sum of 3 means remainder 3, and a negative sum such as $-5$ means $11 - 5 = 6$ .

Rule for 11

(d_0 + d_2 + d_4 + \cdots) - (d_1 + d_3 + \cdots) = 0, \pm 11, \pm 22, \ldots

Worked example

Is 4,893 divisible by 7? Is 90,816 divisible by 11?

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The same idea in a practice question:

Practice questionModerate

Example 3 · Number System · Divisibility Rules

246895

divisible by

11

11 passes as well as 0

An alternating sum of 11 or

-11

means divisible, just as 0 does. Only a result that is not a multiple of 11 means no.

Subtract twice the last digit

For 7 the doubled last digit is taken away, not added: 504 gives

50 - 8 = 42

. Adding gives 58 and the wrong verdict.

Drill 3 more on rules for 7 and 11

Concept 4 of 6: Divisors made of two rules: 6, 12, 15, 18, 36

A number divisible by 3 and by 4 is divisible by 12, because 3 and 4 share no factor except 1. Split the divisor into two such pieces and run both tests.

Definition

$6 = 2 \times 3$ , $12 = 3 \times 4$ , $15 = 3 \times 5$ , $18 = 2 \times 9$ , $24 = 3 \times 8$ , $36 = 4 \times 9$ .
The two pieces must share no factor except 1. $12 = 2 \times 6$ does not work: 18 passes the tests for 2 and 6 and is not divisible by 12.
Both tests must pass. One failure is enough to say no.

Combining two rules

a \mid N,\ b \mid N,\ a \text{ and } b \text{ share no factor} \;\Rightarrow\; ab \mid N

Worked example

Is 7,524 divisible by 36?

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 4 · Number System · Divisibility Rules

Which of these is divisible by

12

The pieces must share nothing

12 is divisible by 4 and by 6 but not by 24. Test 24 as

3 \times 8

, never as

4 \times 6

Drill 4 more on divisors made of two rules: 6, 12, 15, 18, 36

Concept 5 of 6: A missing digit, or the least number to add or take away

Write the rule with the unknown digit as a letter and solve for it; a digit is one of only ten values. To make a number divisible, find its remainder first: adding the divisor minus the remainder finishes the step up, and taking away the remainder steps down.

Definition

Digit sum rules (3, 9): add the known digits, then pick the digit that lifts the total to the next multiple.
Tail rules (4, 8): only the last digits matter. Several digits may work, so test each option.
Rule for 11: write the alternating sum with the letter in it and make it 0 or a multiple of 11.
If $N$ leaves remainder $r$ on division by $d$ : the least number to add is $d - r$ ; the least to subtract is $r$ . For example, $2{,}519 = 9 \times 279 + 8$ , so add 1 or take away 8.

Least number to add or subtract

N = dq + r:\quad \text{add } d - r,\ \text{or subtract } r

Worked example

Find the digit

k

that makes

3k754

divisible by 9.

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Example 5 · Number System · Divisibility Rules

Find digit

m

45m8

to make it divisible by

9

Add the divisor minus the remainder

4,327 leaves remainder 1 on division by 7, so 6 must be added. Adding the remainder itself gives 4,328, which leaves 2.

Drill 3 more on a missing digit, or the least number to add or take away

Concept 6 of 6: Counting and adding the multiples in a range

The multiples of

d

in a range are

d

times a run of whole numbers. Find the first and last of that run and count it; to add the multiples, use first plus last, times the count, halved.

Definition

First multiple of $d$ at or above $a$ , last at or below $b$ : write them as $d \times m$ and $d \times n$ . The count is $n - m + 1$ .
Largest multiple of $d$ up to $N$ : $N$ minus its remainder. The largest 4-digit multiple of 7 is $9{,}999 - 3 = 9{,}996$ .
Sum of the multiples: $\tfrac{\text{count} \times (\text{first} + \text{last})}{2}$ .
Divisible by both $a$ and $b$ : divisible by their LCM, which is $ab$ when they share no factor. (The LCM in general comes on Factors and Multiples.)

Multiples of d from d·m to d·n

\text{count} = n - m + 1,\qquad \text{sum} = \dfrac{\text{count} \times (dm + dn)}{2}

Worked example

How many 3-digit numbers are divisible by 13?

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Example 6 · Number System · Divisibility Rules

How many 3-digit numbers are divisible by both

7

and

11

Count both ends

From

4 \times 1

4 \times 10

there are

10 - 1 + 1 = 10

multiples of 4, not 9. Subtracting the two multipliers forgets one end.

Both 4 and 6 means 12

4 and 6 share the factor 2, so a number divisible by both is a multiple of 12, not of 24. Multiply only when the two share no factor.

Drill 3 more on counting and adding the multiples in a range

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (5)

Rules that add the digits: 3 and 9
Digit-sum rule
$N \text{ divisible by } 9 \;\text{means}\; \text{its digit sum is divisible by } 9$
Rules for 7 and 11
Rule for 11
$(d_0 + d_2 + d_4 + \cdots) - (d_1 + d_3 + \cdots) = 0, \pm 11, \pm 22, \ldots$
Divisors made of two rules: 6, 12, 15, 18, 36
Combining two rules
$a \mid N,\ b \mid N,\ a \text{ and } b \text{ share no factor} \;\Rightarrow\; ab \mid N$
A missing digit, or the least number to add or take away
Least number to add or subtract
$N = dq + r:\quad \text{add } d - r,\ \text{or subtract } r$
Counting and adding the multiples in a range
Multiples of d from d·m to d·n
$\text{count} = n - m + 1,\qquad \text{sum} = \dfrac{\text{count} \times (dm + dn)}{2}$

Reference tables (1)

Rules that read the last digits: 2, 4, 5, 8, 10 and 256 rows

Divisor	Test	Example that passes
2	Last digit is 0, 2, 4, 6 or 8	5,318
4	Last two digits form a multiple of 4	7,316 (16) An even last digit is not enough: 1,326 ends in 26, which is not a multiple of 4.
8	Last three digits form a multiple of 8	41,128 (128)
5	Last digit is 0 or 5	6,035
10	Last digit is 0	4,560
25	Last two digits are 00, 25, 50 or 75	3,175

Watch out for (8)

Two digits are not enough for 8
→ Rules that read the last digits: 2, 4, 5, 8, 10 and 25
Divisible by 3 is not divisible by 9
→ Rules that add the digits: 3 and 9
11 passes as well as 0
→ Rules for 7 and 11
Subtract twice the last digit
→ Rules for 7 and 11
The pieces must share nothing
→ Divisors made of two rules: 6, 12, 15, 18, 36
Add the divisor minus the remainder
→ A missing digit, or the least number to add or take away
Count both ends
→ Counting and adding the multiples in a range
Both 4 and 6 means 12
→ Counting and adding the multiples in a range

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