Tests of divisibility

Doing long division just to know whether $4{,}536$ is divisible by $9$ is a waste. Mathematicians long ago discovered divisibility tricks , quick checks based on the digits of a number that let you answer "divisible or not" in seconds.

Concept

Here are the most useful divisibility rules:

Divisible by $2$ . The number ends in $0, 2, 4, 6$ , or $8$ (i.e. its last digit is even).

Examples: $48$ , $130$ , $9{,}372$ .

Divisible by $3$ . The sum of its digits is divisible by $3$ .

Example: $1{,}245$ . Digit sum $1+2+4+5 = 12$ . $12$ is divisible by $3$ . So $1{,}245$ is divisible by $3$ .

Divisible by $4$ . The last two digits form a number divisible by $4$ .

Example: $5{,}312$ . Last two: $12$ . $12 \div 4 = 3$ . Yes, divisible by $4$ .

Divisible by $5$ . The last digit is $0$ or $5$ .

Examples: $25$ , $130$ , $4{,}075$ .

Divisible by $6$ . Divisible by both $2$ and $3$ .

Example: $1{,}242$ . Even ✓. Digit sum $9$ , divisible by $3$ ✓. So divisible by $6$ .

Divisible by $9$ . The sum of its digits is divisible by $9$ .

Example: $1{,}458$ . Digit sum $1+4+5+8 = 18$ . $18 \div 9 = 2$ . Yes.

Divisible by $10$ . The last digit is $0$ .

Examples: $30$ , $1{,}000$ , $73{,}450$ .

Divisible by $11$ . Take the alternating sum of digits (from the right or left). If that is divisible by $11$ (including $0$ ), the number is divisible by $11$ .

Example: $2{,}728$ . Alternating sum: $2 - 7 + 2 - 8 = -11$ . Divisible by $11$ ✓.

Why these rules work

For divisibility by $9$ : notice that $10 = 9 + 1$ , $100 = 99 + 1$ , $1000 = 999 + 1$ , and so on. So a number like $\overline{abcd} = 1000a + 100b + 10c + d = (999a + 99b + 9c) + (a + b + c + d)$ . The first part is always a multiple of $9$ . So the whole number is divisible by $9$ exactly when $a + b + c + d$ is. Same idea works for $3$ since $9$ is a multiple of $3$ .

Divisibility tests turn a "calculate" question into a "look" question, and they save real time when you are working with prime factorisation, fractions, or any factor-finding work.

Worked examples

Example 1. Is $5{,}823$ divisible by $3$ ? By $9$ ?

Digit sum $= 5+8+2+3 = 18$ . Divisible by both $3$ and $9$ .
So yes to both.

Example 2. Is $1{,}234$ divisible by $4$ ?

Last two digits: $34$ . $34 \div 4 = 8.5$ .
Not divisible by $4$ .

Example 3. Is $9{,}900$ divisible by $11$ ?

Alternating sum (right to left): $0 - 0 + 9 - 9 = 0$ . Yes, divisible by $11$ .

Example 4. Find the smallest digit $\#$ such that $34{,}\#82$ is divisible by $3$ .

Digit sum: $3 + 4 + \# + 8 + 2 = 17 + \#$ .
Need $17 + \#$ divisible by $3$ . Smallest non-negative $\#$ that works: $17 + 1 = 18$ ✓.
So $\# = 1$ . The number $34{,}182$ is divisible by $3$ .

Example 5. Is $24{,}680$ divisible by $5$ and by $10$ ?

Last digit $0$ , so divisible by both $5$ and $10$ .

Try it yourself

Is $7{,}776$ divisible by $2$ ? By $3$ ? By $9$ ?
Is $3{,}420$ divisible by $6$ ?
Is $98$ divisible by $4$ ? By $11$ ?
Find the smallest digit $\#$ such that $5{,}3\#4$ is divisible by $4$ .
Find the smallest digit $\#$ such that $\#65$ is divisible by $3$ .
Is $13{,}486$ divisible by $9$ ? If not, by what is it short?
Why do divisibility tests for $3$ and $9$ both use the digit sum?
Investigate: design a divisibility test for $25$ . (Hint: look at the last two digits.)

Activity

Phone number game. Pick any $10$ -digit phone number (yours or a friend's). Test it for divisibility by $2$ , $3$ , $5$ , $9$ , and $11$ using the rules above. Write down which divisibility tests it passes. Repeat with two more phone numbers. Are any of them prime? (Hint: a number divisible by $2$ and not equal to $2$ cannot be prime.)