Question 1

Show that the number 132 is not perfect.

Accepted Answer

The proper divisors

Question 2

Show that the numbers 140 and 165 are not amicable.

Accepted Answer

The proper divisors of 140 are

Question 3

Use divisibility tests to decide whether the first number is divisible by the second.
-6,955,200; 7 [Note that a divisibility test for 7 is as follows:
Double the last digit of the number and subtract this value from the original number with the last
Digit omitted. Repeat this process as many times as necessary until the number obtained can easily
Be divided by 7. If the final number obtained is divisible by 7, then so is the original number. If the
Final number obtained is not divisible by 7, then neither is the original number.]

Accepted Answer

A) Yes 
B) No 
A) Yes 
B) No

Question 4

Find the number of divisors of the number.
-$2 ^ { 5 } \cdot 3 ^ { 6 } \cdot 5 ^ { 2 }$

Accepted Answer

A)  120 
B)  126 
C)  60 
D)  90 
A)  120 
B)  126 
C)  60 
D)  90

Question 5

Find the greatest common factor of the numbers in the group.
-42, 56, 98

Accepted Answer

A) 7 
B) 2 
C) 14 
D) 28 
A) 7 
B) 2 
C) 14 
D) 28

Question 6

Use divisibility tests to decide whether the first number is divisible by the second.
-404,036; 4

Accepted Answer

A) Yes 
B) No 
A) Yes 
B) No

Question 7

Determine whether the statement is true or false.
-Two composite numbers can never be relatively prime.

Accepted Answer

A) True 
 B)False

Question 8

Determine whether or not one or more pairs of twin primes exist between the pair of numbers given. If so, identify the twin primes.
-80 and 100

Accepted Answer

A) 89, 91 
B) 83, 85 
C) No 
D) 91, 93 
A) 89, 91 
B) 83, 85 
C) No 
D) 91, 93

Question 9

Determine whether or not one or more pairs of twin primes exist between the pair of numbers given. If so, identify the twin primes.
-34 and 49

Accepted Answer

A) No 
B) 43, 47 
C) 41, 42 and 47, 49 
D) 41, 43 
A) No 
B) 43, 47 
C) 41, 42 and 47, 49 
D) 41, 43

Question 10

Determine whether the number is abundant or deficient.
-108

Accepted Answer

A) Abundant 
B) Deficient 
A) Abundant 
B) Deficient

Question 11

Determine whether or not one or more pairs of twin primes exist between the pair of numbers given. If so, identify the twin primes.
-4 and 16

Accepted Answer

A) 5, 7 and 11, 13 
B) 5, 7 and 9, 11 
C) 5, 7 
D) No 
A) 5, 7 and 11, 13 
B) 5, 7 and 9, 11 
C) 5, 7 
D) No

Question 12

Find the least common multiple of the numbers in the group.
-84, 126

Accepted Answer

A) 252 
B) 42 
C) 504 
D) 756 
A) 252 
B) 42 
C) 504 
D) 756

Question 13

Determine whether the statement is true or false.
-For any prime number $\mathrm { p }$ except 2 , either $\mathrm { F } _ { \mathrm { p } } + 1$ or $\mathrm { F } _ { \mathrm { p } - 1 }$ is divisible by $\mathrm { p } . \mathrm { F } _ { \mathrm { n } }$ represents the nth term of the Fibonacci sequence.

Accepted Answer

A) True 
 B)False

Question 14

Determine whether the statement is true or false.
-The proper divisors of a natural number include all divisors of the number except 1.

Accepted Answer

A) True 
 B)False

Question 15

Find the greatest common factor of the numbers in the group.
-1562, 450, 6750

Accepted Answer

A) 2 
B) 5 
C) 12 
D) 4 
A) 2 
B) 5 
C) 12 
D) 4

Question 16

$\begin{array} { l } 
\frac { 1 } { 1 } = 1 \
\frac { 2 } { 1 } = 2 \
\frac { 3 } { 2 } = 1.5
\end{array}$

Accepted Answer

A)  $\frac { 3 } { 1 } = 3$ 
B)  $\frac { 2 } { 3 } = 0.666 \ldots$ 
C)  $\frac { 8 } { 5 } = 1.6$ 
D)  $\frac { 5 } { 3 } = 1.666 \ldots$ 
A)  $\frac { 3 } { 1 } = 3$ 
B)  $\frac { 2 } { 3 } = 0.666 \ldots$ 
C)  $\frac { 8 } { 5 } = 1.6$ 
D)  $\frac { 5 } { 3 } = 1.666 \ldots$

Question 17

Solve the problem relating to the Fibonacci sequence.
-$F _ { 25 } = 75,025 , F _ { 26 } = 121,393$
Find $\mathrm { F } _ { 27 }$.

Accepted Answer

A)  $\mathrm { F } _ { 27 } = 196,418$ 
B)  $\mathrm { F } _ { 27 } = 242,786$ 
C)  $\mathrm { F } _ { 27 } = 167,761$ 
D)  $\mathrm { F } _ { 27 } = 46,368$ 
A)  $\mathrm { F } _ { 27 } = 196,418$ 
B)  $\mathrm { F } _ { 27 } = 242,786$ 
C)  $\mathrm { F } _ { 27 } = 167,761$ 
D)  $\mathrm { F } _ { 27 } = 46,368$

Question 18

Use the formula indicated to determine the prime number generated for the given value of n.
-$p = n ^ { 2 } - 79 n + 1,601 ; n = 65$

Accepted Answer

A) 7759 
B) 4201 
C) 10,961 
D) 691 
A) 7759 
B) 4201 
C) 10,961 
D) 691

Question 19

Determine whether the number is abundant or deficient.
-18

Accepted Answer

A) Abundant 
B) Deficient 
A) Abundant 
B) Deficient

Question 20

Use the Lucas sequence to answer the question.
-What is the 26th term of the Lucas sequence?

Accepted Answer

A) 165,061 
B) 167,761 
C) 103,682 
D) 271,443 
A) 165,061 
B) 167,761 
C) 103,682 
D) 271,443

Show that the number 132 is not perfect.

Show that the numbers 140 and 165 are not amicable.

Find the number of divisors of the number. - $2 ^ { 5 } \cdot 3 ^ { 6 } \cdot 5 ^ { 2 }$

Find the greatest common factor of the numbers in the group. -42, 56, 98

Use divisibility tests to decide whether the first number is divisible by the second. -404,036; 4

Determine whether the statement is true or false. -Two composite numbers can never be relatively prime.

Determine whether or not one or more pairs of twin primes exist between the pair of numbers given. If so, identify the twin primes. -80 and 100

Determine whether or not one or more pairs of twin primes exist between the pair of numbers given. If so, identify the twin primes. -34 and 49

Determine whether the number is abundant or deficient. -108

Determine whether or not one or more pairs of twin primes exist between the pair of numbers given. If so, identify the twin primes. -4 and 16

Find the least common multiple of the numbers in the group. -84, 126

Determine whether the statement is true or false. -The proper divisors of a natural number include all divisors of the number except 1.

Find the greatest common factor of the numbers in the group. -1562, 450, 6750

=1 =2 =1.5

Solve the problem relating to the Fibonacci sequence. - $F _ { 25 } = 75,025 , F _ { 26 } = 121,393$ Find $\mathrm { F } _ { 27 }$ .

Use the formula indicated to determine the prime number generated for the given value of n. - $p = n ^ { 2 } - 79 n + 1,601 ; n = 65$

Determine whether the number is abundant or deficient. -18

Use the Lucas sequence to answer the question. -What is the 26th term of the Lucas sequence?

The Art of Problem Solving

The Basic Concepts of Set Theory

Introduction to Logic

Numeration Systems

The Real Numbers and Their Representations

The Basic Concepts of Algebra

Graphs, Functions, and Systems of Equations and Inequalities

Geometry

Counting Methods

Probability

Statistics

Personal Financial Management

Trigonometry Formerly

Graph Theory

Voting and Apportionment

Filters

Exam 5: Number Theory

Show that the number 132 is not perfect.

Show that the numbers 140 and 165 are not amicable.

Find the number of divisors of the number. - 25⋅36⋅522 ^ { 5 } \cdot 3 ^ { 6 } \cdot 5 ^ { 2 }25⋅36⋅52

Find the greatest common factor of the numbers in the group. -42, 56, 98

Use divisibility tests to decide whether the first number is divisible by the second. -404,036; 4

Determine whether the statement is true or false. -Two composite numbers can never be relatively prime.

Determine whether or not one or more pairs of twin primes exist between the pair of numbers given. If so, identify the twin primes. -80 and 100

Determine whether or not one or more pairs of twin primes exist between the pair of numbers given. If so, identify the twin primes. -34 and 49

Determine whether the number is abundant or deficient. -108

Determine whether or not one or more pairs of twin primes exist between the pair of numbers given. If so, identify the twin primes. -4 and 16

Find the least common multiple of the numbers in the group. -84, 126

Determine whether the statement is true or false. -The proper divisors of a natural number include all divisors of the number except 1.

Find the greatest common factor of the numbers in the group. -1562, 450, 6750

=1 =2 =1.5

Solve the problem relating to the Fibonacci sequence. - F25=75,025,F26=121,393F _ { 25 } = 75,025 , F _ { 26 } = 121,393F25​=75,025,F26​=121,393 Find F27\mathrm { F } _ { 27 }F27​ .

Use the formula indicated to determine the prime number generated for the given value of n. - p=n2−79n+1,601;n=65p = n ^ { 2 } - 79 n + 1,601 ; n = 65p=n2−79n+1,601;n=65

Determine whether the number is abundant or deficient. -18

Use the Lucas sequence to answer the question. -What is the 26th term of the Lucas sequence?

The Art of Problem Solving

The Basic Concepts of Set Theory

Introduction to Logic

Numeration Systems

The Real Numbers and Their Representations

The Basic Concepts of Algebra

Graphs, Functions, and Systems of Equations and Inequalities

Geometry

Counting Methods

Probability

Statistics

Personal Financial Management

Trigonometry Formerly

Graph Theory

Voting and Apportionment

Filters

Find the number of divisors of the number. - $2 ^ { 5 } \cdot 3 ^ { 6 } \cdot 5 ^ { 2 }$

Solve the problem relating to the Fibonacci sequence. - $F _ { 25 } = 75,025 , F _ { 26 } = 121,393$ Find $\mathrm { F } _ { 27 }$ .

Use the formula indicated to determine the prime number generated for the given value of n. - $p = n ^ { 2 } - 79 n + 1,601 ; n = 65$