Download CHAPTER 2 NUMBERS AND OPERATIONS ON NUMBERS

CHAPTER 3 FACTORS AND MULTIPLES
Factors and multiples 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 variety of SAT Problems. You will see this
as you review the 2 Solved SAT Problems, complete the 15 Practice SAT Questions and review the answer
explanations.
FACIORS
A factor of a whole number divides the number, with no remainder.
Example:
Is 6 a factor of 50?
8
6 50
48 Because 6 divides into 50 eight times with remainder 2, 6 is not a factor of 50.
2
The answer to the division problem above can be rewritten as 8
2
1
=8 =8.333
6
3
Example:
Is 7 a factor of 63?
9
7 63
63 Because 7 divides into 63 nine times without a remainder, 7 is a factor of 63.
0
PRIMES AND COMPOSITES
A prime number is a positive integer that has exactly two factors, 1 and itself.
A composite is a positive integer that has more than two 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 arc 1,17 . Therefore, 17 is a prime number.
The GREATEST COMMON FACTOR (GCF) of two numbers is the largest factor the two 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.
PRIME FACTORIZATION is a positive integer written as a product of prime numbers.
Example:
What is the prime factorization of 72?
72  23  32
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 .
The LEAST COMMON MULTIPLE (LCM) is the smallest multiple two numbers have in common.
Example:
What is the LCM of 3 and 6?
Answer: 6
Look at the list of multiples shown above.
Notice that 6 is the smallest multiple these numbers have in common.
Practice Questions
1.
List the factors of the following numbers and then identify the greatest common factor.
(A) 34
2.
State whether the following numbers are prime or composite.
(A) 31
3.
(B) 45
List the first eight multiples of each number and then identify the least common multiple of the two
(A) 5
Practice Answers
(A) 1, 2,17,34
(B) 1, 2, 4,13, 26,52
The GCF is 2.
2.
(A) Prime
(B) Composite
(C) Composite
(D) Prime
3.
(D) 59
(B) 195
numbers.
1.
(C) 63
Write the prime factorization for each number.
(A) 44
4.
(B) 52
(A) 22 11
(B) 3  5 13
(B) 7
4.
(A) 5,10,15, 20, 25,30,35, 40
(B) 7,14, 21, 28,35, 42, 49,56
The least common multiple is 35.
SOLVED SAT PROBLEMS
1.
What is the LCM of the numbers that satisfy these conditions?
factor of 48
multiple of 8
Answer: 8
The factors of 48 are 1, 2, 3, 4, 6, 8,12,16, 24, 48 . Of these numbers 8,16, 24, 48 are also multiples of 8.
8 is the LCM of these numbers.
2.
If a, b, and c are distinct integers such that a  b  c , b  0 , and c  a  b , then which of the following
must be true?
(A) c is a positive number.
(B) a is a prime number.
(C) b is a prime number.
(D) c is a factor of a.
(E) c is a prime number.
Answer: A
Because a  b  c , b  0 , both a and b are negative integers. c  a  b so c is a positive integer. The product of two negative
integers is a positive integer.
PRACTICE SAT QUESTIONS
（
）1. What is the sum of all the factors of 24?
(A) 46
(B) 49
(C) 50
(D) 60
(E) 66
Answer: D
The factors of 24 are 1, 2, 3, 4, 6, 8,12, 24 .
（
1  2  3  4  6  8  12  24  60
）2. 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
Answer: C
3  3  3  3  3  3  243
5
（
）3. 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.
(D) It is a multiple of 5.
(E) It is a factor of 10.
(C) It is a factor of 156.
Answer: C
23 and 29 are the only prime numbers between 20 and 30.
23  29  52
52  3  156
This shows that 52 is a factor of 156.
（
）4. At a dinner party each table can seat eight people. If 100 people attend the party, what is the
minimum number of tables that are needed?
(A) 12
(B) 13
(C) 14
(D) 15
(E) 16
Answer: B
Divide the number of people at the party by the number of seats around each table, 100  8  12.5
Thirteen tables will be needed to seat all 100 people.
（
）5. What is the greatest integer that evenly divides both 48 and 64?
Answer: 16
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, 8,12,16, 32, 64 .The GCF is 16.
（
）6. Let P be a prime number greater than 4. How many distinct prime factors does 9  P have?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6
Answer: A
Because P is a prime number, the prime factorization of 9  P is 3  P . Therefore, 3 and P are the two distinct
2
primes of 9  P .
（
）7. When x is divided by 8, the remainder is 3. What is the remainder when 4 x is divided by 8?
Answer: D
Try an example. Choose 3 for x.
Multiply 3  4  12 .
3 divided by 8 leaves a remainder of 3.
12 divided by 8 leaves a remainder of 4.
（
）8. Let x, y, and z be positive integers such that y is a multiple of x and z. All of the following
statements are true except for:
(C) x  z is a factor of y.
(A) y  z is a multiple of x.
(B) x is a factor of y.
(D) z divides evenly into y.
(E) x divides evenly into y  z .
Answer: C
Note that this item asks for the choice that is NOT true.
We know y is a multiple of x and z. That means y times any number is also a multiple of x, so choice A must be
true. Also, because we know y is a multiple of x and z, this means that x and z divide evenly into y, so choices B,
D and E are true.
The process of elimination leaves choice C.
Use an example to check this. Let x  3 , y  12 , and z  6 . y is a multiple of x and z. but x  z  18 is not a
multiple of y  12 .
（
）9. When a two-digit number is divided by 5 the remainder is 2. Which of the following statements
must be true about the two-digit number?
(A) The sum of all the digits is odd.
(B) The digit in the one’s place is odd.
(C) The number is prime.
(D) The number is odd.
(E) The digit in the one’s place is prime.
Answer: E
The two-digit numbers that have a remainder of 2 when divided by 5 are
12,17, 22, 27, 32, 37, 42, 47, 52, 57, 62, 67, 72, 77, 82, 87, 92, 97
Only choice E is always true, the one’s digit is prime.
Don’t be fooled because some of the answers are true sometimes.
For cxample, in choice (A) the sum of all digits is odd.
This is true if the two-digit number is 32.
The sum of the digits 3  2  5 is odd.
But choice (A) is not always true.
If the two digit number is 42, the sum of the digits 4  2  6 is not odd.
（
）10. The sum of 3 consecutive integers is 15. How many distinct prime factors does the product of
these three numbers have?
(A) 2
(B) 3
(C) 4
(D) 5
Answer: B
The numbers must be 4, 5, and 6.
4  5  6  15 and
2,3, and 5 are the three distinct pnme factors.
4  5  6  120  2  3  5
3
(E) 6
（
）11. If a, b, and c are all integers greater than 1 and a  b  21 and b  c  39 , then which of the
following choices gives the correct ordering of the numbers?
(A) b  a  c
(B) c  a  b
(C) a  b  c
(D) b  c  a
(E) a  c  b
Answer: A
The question states that a, b, and c are integers greater than 1. That means b is an integer greater than 1 that
divides evenly into both 21 and 39. That means that b must be 3.
Divide. a must be 7, 7  3  21 . c must be 13, 3  13  39 , 3  7  13 , b  a  c
（
）12.How many positive integers less than 20 have an odd number of distinct factors?
(A) 12
(B) 10
(C) 8
(D) 6
(E) 4
Answer: E
The only positive integers less than 20 that have an odd number of factors are the squares 1,4,9, and 16.
That is because these numbers, alone, have a factor that is multiplied by itself.
The factors for each are as follows:
（
Number
Facors
1:
1
Number
Facors
9:
1, 3, 9
Number
4:
Number
16:
Facors
1, 2, 4
Facors
1, 2, 4, 8,16
）13. In the repeating decimal 0.714285714285…, what is the 50th digit to the right of the decimal
point?
Answer: 1
In the repeating dccimal 0.714285714285. . . , 5 is the 6th, 12th. 18th. 24th, 30th, 36th. 42nd, and 48th digit.
7 is the 49th digit and 1 is the 50th digit.
You might also realize that there are 6 digits that repeat, divide 50 by 6, and get a remainder of 2.
Therefore the 2nd of the 6 digits that repeat, which is 1, will be the 50th digit.
（
）14. How many distinct composite numbers can be formed by adding 2 of the first 5 prime numbers ?
Answer: 7
The first 5 prime numbers are 2, 3, 5, 7,11 .
Add pairs of primes
5,7,and 11 are prime numbers.
8,9,10,12,14,16, and 18 are composite numbers.
235
25  7
27 9
2  9  11
35  8
3  7  10
3  11  14
5  7  12
5  11  16
7  11  18
（
）15. A rope is 13 feet long. How many ways can the rope be cut into more than one piece so that the
length of each piece is a prime number?
(A) 4
(B) 5
(C) 6
Answer: E
Try a diagram.
The different possibilities are
2,2,2,2,2,3
█─┼─█─┼─█─┼─█─┼─█─┼─█─┼─┼─█
2,2,2,2,5
█─┼─█─┼─█─┼─█─┼─█─┼─┼─┼─┼─█
2,2,2,7
█─┼─█─┼─█─┼─█─┼─┼─┼─┼─┼─┼─█
2,2,3,3,3
█─┼─█─┼─█─┼─┼─█─┼─┼─█─┼─┼─█
2,11
█─┼─█─┼─┼─┼─┼─┼─┼─┼─┼─┼─┼─█
2,3,3,5
█─┼─█─┼─┼─█─┼─┼─█─┼─┼─┼─┼─█
3,3,7
█─┼─┼─█─┼─┼─█─┼─┼─┼─┼─┼─┼─█
3,5,5
█─┼─┼─█─┼─┼─┼─┼─█─┼─┼─┼─┼─█
(D) 7
(E) 8