Divisibility

Learnto use divisibilityrules.

This year, 42 girls signed up to playbasketball for the Junior Girls League,which has 6 teams. To find whethereach team can have the same numberof girls, decide if 42 is divisible by 6.

A number is divisible by anothernumber if the quotient is a wholenumber with no remainder.

Vocabularydivisible

composite number

prime number

42 + 6 =7 ~ Quotient

Since there is no remainder, 42 is divisibleby 6.The Junior GirlsLeaguecan have 6 teams with 7 girls each.

Divisibility Rules

NotDivisible I DivisibleA number is divisible by...

2 if the last digit is even (0, 2, 4, 6, or 8).

3 if the sum of the digits is divisible by 3.

4 if the last two digits form a number divisible by 4.

5 if the last digit is 0 or 5.

6 if the number is divisible by both 2 and 3.

9 if the sum of the digits is divisible by 9.

10 if the last digit is O.

EXAMPLE Checking Divisibility

\.A> Tellwhether 610 is divisible by 2,3,4, and 5.

3,978

315

8,512

14,975

48

711

15,990

,4,975

139

7,518

10,978

20

93

10,536

So 610 is divisible by 2 and 5.

152 Chapter 4 Number Theory and Fractions

2 I The last digit, 0, is even. I Divisible

3 I The sum of the digits is 6 + 1 + 0 = 7. I Not divisible

7 is not divisible by 3.

4 I The last two digits form the number 10. I Not divisible

10 is not divisible by 4.

5 The last digit is O. Divisible

,) Tell whether 387 is divisible by 6, 9, and 10.

6 The last digit, 7, is odd, so 387 is notdivisible by 2.

The sum of the digits is 3 + 8 + 7 = 18.

18 is divisible by 9.

The last digit is 7, not o. Not divisible

Not divisible

9Divisible

10

So 387 is divisible by 9.

Any number greater than 1 is divisible by at least two numbers-1 and the number itself. Numbers that are divisible by more than twonumbers are called composite numbers.

A prime number is divisible by only the numbers 1 and itself. Forexample, 11 is a prime number because it is divisible by only 1 and 11.The numbers 0 and 1 are neither prime nor composite.

EXAMP LE ~ Identifying Prime and Composite Numbers

Tellwhether each number is prime or composite.

~45divisible by 1, 3, 5, 9, 15,45composite

J 13

divisible by 1, 13

prime

,,; 19

divisible by 1, 19

prime

'--.J 49divisible by 1, 7, 49composite~

The prime numbers from 1 through 50 are highlighted below.

4-1 Divisibility 153

-- -

2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50 I

f' -Think and Discuss

1. Tell which whole numbers are divisible by 1.

2. Explain how you know that 87 is a composite number.

3. Tell how the divisibilityrules help you identify composite numbers.