Factors,  Mul-ples  And  Divisibility  

Introduction:�Factors  multiples  and  divisibilit3  deal  with  dividing  and  multiplying  positive  integers  {1,2,3,4,  .  .  .}.  In  this  chapter  you  will  work  with  such  concepts  as  Greatest  Common  Factor  (GCF)  and  Least  Common  Multiple  (LCM).  You  will  use  the  factors  and  multiples  of  a  number  to  help  you  solve  a  variet3  of  SAT  Problems.    

Factors:�A  factor  of  a  whole  number  divides  the  number,  with  no  remainder.      Example:      

Is  6  a  factor  of  50?    Because  6  divides  into  50  eight  times  with  remainder  2,  6  is  not  a  factor  of  50.      Example:      

Is  7  a  factor  of  63?    Because  7  divides  into  63  nine  times  without  a  remainder,  7  is  a  factor  of  63.      

PRIMES AND COMPOSITES �A  prime  number  is  a  positive  integer  that  has  exactly  t6o  factors,  1  and  itself  A  composite  is  a  positive  integer  that  has  more  than  t6o  factors      Example:    Is  20  a  prime  or  composite  number?    The  factors  of  20  are  {1,2,4,5,10,20}.  Therefore,  20  is  a  composite  number.      Example:    Is  17  a  prime  or  composite  number?  The  factors  of  17  are  {1,17}.  Therefore,  17  is  a  prime  number.      

GREATEST COMMON FACTOR (GCF) �The  GREATEST  COMMON  FACTOR  (GCF)  of  t6o  numbers  is  the  largest  factor  the  t6o  numbers  have  in  common.      Example:      What  is  the  GCF  of  16  and  36?      The  factors  of  16  are  {1,2,4,8,16},  and  the  factors  of  36  are  {1,2,3,4,6,9,12,  18,36}.      Therefore,  the  GCF  is  4.      

MULTIPLES �The  MULTIPLES  of  a  given  number  are  those  numbers  created  by  successive  multiplication.  The  given  number  divides  the  multiple  without  a  remainder.      Example:      List  the  first  five  multiples  of  3  and  the  first  five  multiples  of  6.    The  multiples  of  3  are  {3,6,9,12,15},  and  the  multiples  of  6  are  {6,12,18,  24,30}.      

LEAST COMMON MULTIPLE (LCM) �The  LEAST  COMMON  MULTIPLE  (LCM)  is  the  smallest  multiple  t6o  numbers  have  in  common.      Example:      What  is  the  LCM  of  3  and  6?  Answer:  6    6  is  the  smallest  multiple  these  numbers  have  in  common.      

Divisibility:�If  you  divide  one  number  by  another,  the  result  is  a  whole  number  without  a  remainder.    Example:    12  /  6  =2  No  remainder,  so  12  is  divisible  by  6.    

Practice Questions�

What is the sum of all the factors of 24?���

A. 46

B. 49

C. 50

D. 60

E. 66

What is the greatest number of 3s that can be multiplied together and still

have a result less than 250?��� A. 3

B. 4

C. 5

D. 6

E. 7

Which of the following must be true about the sum of all the prime numbers

between 20 and 30? ������ A.  It is a prime number.

B.  It is an odd number.

C.  It is a factor of 156.

D.  It is a multiple of 5.

E.  It is a factor of 10.

What is the greatest integer that evenly divides both 48 and 64? The largest integer that evenly divides both

48 and 64 is the GCF of the two numbers.

The factors of 48 are

{1,2,3,4,6,8,12,16,24,48}.

The factors of 64 are {1,2,4,8,16,32,64}.

The GCF is 16.

Answer: 16

When x is divided by 8, the remainder is 3. What is the remainder when 4x is

divided by 8?���

Try an example.

Choose 11 for x.

11 divided by 8 leaves a remainder of 3.

Multiply 11 × 4 = 44.

44 divided by 8 leaves a remainder of 4.

Answer: 4