Divisibility Intuition Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Divisibility Intuition.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Understanding when one whole number divides evenly into another, leaving no remainder—the foundation of factor and multiple relationships.

Can you share 12 cookies equally among 4 people? Yes, 3 each. 12 is divisible by 4.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: $b$ divides $a$ when $a$ splits into equal groups of $b$ with nothing left over.

Common stuck point: The procedure for divisibility intuition is the easy part; the trap is writing the divides bar backwards. Asking "Does the larger number split into equal whole groups of the smaller with nothing left over?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the larger number split into equal whole groups of the smaller with nothing left over?

Worked Examples

Example 1

easy

Use divisibility rules to determine whether

4{,}836

is divisible by

2

3

4

6

, and

9

Answer

4{,}836

is divisible by

2, 3, 4, 6

but not by

9

First step

2

: last digit is

6

(even). Yes.

Full solution

2
By $3$ : digit sum $= 4+8+3+6 = 21$ ; $21 \div 3 = 7$ . Yes.
3
By $4$ : last two digits $36$ ; $36 \div 4 = 9$ . Yes.
4
By $6$ : divisible by both $2$ and $3$ . Yes.
5
By $9$ : digit sum $21$ ; $21 \div 9 = 2.33\ldots$ Not a whole number. No.

Divisibility rules are shortcuts derived from properties of our base-

10

system. The rules for

2

and

5

check the last digit; for

3

and

9

, sum the digits; for

4

, check the last two digits. These avoid long division for quick classification.

Example 2

medium

Explain why the divisibility rule for

3

works: a number is divisible by

3

if and only if the sum of its digits is divisible by

3

Example 3

medium

A whole number is divisible by

15

exactly when it is divisible by both ___ and ___. Fill in.

Example 4

hard

Prove the divisibility-by-9 rule: a number is divisible by 9 iff its digit sum is.

Example 5

challenge

Show that the product of any four consecutive integers is divisible by

24

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Test

7{,}215

for divisibility by

5

9

, and

10

using divisibility rules.

Example 2

medium

A number

N

leaves remainder

2

when divided by

5

and remainder

1

when divided by

3

. What are the possible last digits of

N

, and is

N

divisible by

15

Example 3

easy

12

divisible by

4

Example 4

easy

35

divisible by

5

Example 5

easy

Use the digit-sum rule: is

123

divisible by

3

Example 6

easy

90

divisible by

10

Example 7

easy

7

divisible by

2

Example 8

easy

Use the digit-sum rule: is

81

divisible by

9

Example 9

easy

Find a factor of

15

other than

1

and

15

Example 10

easy

24

divisible by

6

Example 11

medium

246

divisible by

6

? Check both conditions.

Example 12

medium

14

divisible by

6

? Show why the

2

-and-

3

check matters.

Example 13

medium

Contrast the rules: is

24

divisible by

3

and by

9

? Use digit sums.

Example 14

medium

Find all single-digit factors of

36

Example 15

medium

Why does the last-digit test work for

2,5,10

but not for

3

Example 16

medium

1{,}000{,}000

divisible by

8

? Use the last-three-digits rule.

Example 17

medium

If a number is divisible by

4

and by

9

, is it divisible by

36

? Explain.

Example 18

medium

7{,}425

divisible by

9

? Use the digit-sum rule.

Example 19

medium

105

divisible by

15

? Check the

3

and

5

conditions.

Example 20

challenge

Prove the divisibility-by-

3

rule: a number is divisible by

3

iff its digit sum is.

Example 21

challenge

Show that if

a\mid b

and

b\mid c

then

a\mid c

Example 22

challenge

Prove that among any three consecutive integers, exactly one is divisible by

3

Example 23

easy

48

divisible by

8

Example 24

easy

99

divisible by

9

? Use the digit-sum rule.

Example 25

easy

250

divisible by

5

? By

10

Example 26

easy

74

divisible by

2

Example 27

easy

List all the single-digit numbers that

60

is divisible by.

Example 28

easy

144

divisible by

4

Example 29

easy

217

divisible by

7

? Use a direct check.

Example 30

medium

312

divisible by

6

? Check both required conditions.

Example 31

medium

5{,}832

divisible by

9

? Use the digit-sum rule.

Example 32

medium

Find the smallest positive integer divisible by both

4

and

6

Example 33

medium

1{,}008

divisible by

8

? Use the last-three-digits rule.

Example 34

medium

2{,}520

divisible by

12

? Check both

3

and

4

Example 35

medium

A number ends in

5

. Could it be divisible by

4

Example 36

medium

n

is divisible by

6

, must

n

be divisible by

12

Example 37

medium

Find the smallest digit

d

so that

13{,}d5

is divisible by

9

Example 38

hard

Find all digits

d

so that

52{,}d34

is divisible by

3

Example 39

hard

a

and

b

are both divisible by

7

, prove that

a - b

is divisible by

7

Example 40

hard

12{,}345{,}678

divisible by

11

? Use the alternating-sum rule.

Example 41

hard

A number

N

is divisible by

18

. List all of the following that

N

must be divisible by:

2, 3, 4, 6, 9, 12

Example 42

hard

Find the smallest positive integer that is divisible by

1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Example 43

challenge

Find the smallest five-digit number divisible by every integer from

1

8

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Related Concepts

Division Factors Multiples Prime Numbers

Background Knowledge

These ideas may be useful before you work through the harder examples.

division