Question 1

$\begin{array} { l } 
2 ^ { 2 } - 1 ^ { 2 } = 3 \
3 ^ { 2 } - 1 ^ { 2 } = 8 \
5 ^ { 2 } - 2 ^ { 2 } = 21
\end{array}$

Accepted Answer

A)  $8 ^ { 2 } - 5 ^ { 2 } = 39$ 
B)  $13 ^ { 2 } - 8 ^ { 2 } = 55$ 
C)  $8 ^ { 2 } - 3 ^ { 2 } = 55$ 
D)  $8 ^ { 2 } - 2 ^ { 2 } = 60$ 
A)  $8 ^ { 2 } - 5 ^ { 2 } = 39$ 
B)  $13 ^ { 2 } - 8 ^ { 2 } = 55$ 
C)  $8 ^ { 2 } - 3 ^ { 2 } = 55$ 
D)  $8 ^ { 2 } - 2 ^ { 2 } = 60$

Question 2

Find the number of divisors of the number.
-60

Accepted Answer

A) 14 
B) 12 
C) 16 
D) 10 
A) 14 
B) 12 
C) 16 
D) 10

Question 3

$\begin{array} { c } 
1 = 2 - 1 \
1 + 1 = 3 - 1 \
1 + 1 + 2 = 5 - 1
\end{array}$

Accepted Answer

A)  $1 + 1 + 2 + 3 = 13 - 1$ 
B)  $1 + 1 + 2 + 3 + 5 = 13 - 1$ 
C)  $1 + 1 + 2 + 2 = 7 - 1$ 
D)  $1 + 1 + 2 + 3 = 8 - 1$ 
A)  $1 + 1 + 2 + 3 = 13 - 1$ 
B)  $1 + 1 + 2 + 3 + 5 = 13 - 1$ 
C)  $1 + 1 + 2 + 2 = 7 - 1$ 
D)  $1 + 1 + 2 + 3 = 8 - 1$

Question 4

Determine whether the statement is true or false.
-If a > b > c, then the least common multiple of paqb, pbqc, and pcqa is paqa.

Accepted Answer

A) True 
 B)False

Question 5

Find all natural number factors of the number.
-42

Accepted Answer

A) 1, 2, 3, 7, 6, 14, 21, 42 
B) 1, 2, 3, 7, 6, 14, 28, 42 
C) 1, 7, 42 
D) 7, 6, 14, 42 
A) 1, 2, 3, 7, 6, 14, 21, 42 
B) 1, 2, 3, 7, 6, 14, 28, 42 
C) 1, 7, 42 
D) 7, 6, 14, 42

Question 6

Answer the question.
-Planets A, B, and C orbit a certain star once every 3, 7, and 18 months, respectively. If the three planets are now in the same straight line, what is the smallest number of months that must pass
Before they line up again?

Accepted Answer

A) 378 months 
B) 28 months 
C) 126 months 
D) 54 months 
A) 378 months 
B) 28 months 
C) 126 months 
D) 54 months

Question 7

Find all natural number factors of the number.
-125

Accepted Answer

A) 1, 5, 25 
B) 1, 5, 25, 125 
C) 5, 62, 125 
D) 5, 25, 125 
A) 1, 5, 25 
B) 1, 5, 25, 125 
C) 5, 62, 125 
D) 5, 25, 125

Question 8

$\begin{array} { l } 
2 ^ { 3 } + 1 ^ { 3 } - 1 ^ { 3 } = 8 \
3 ^ { 3 } + 2 ^ { 3 } - 1 ^ { 3 } = 34
\end{array}$

Accepted Answer

A)  $5 ^ { 3 } + 2 ^ { 3 } - 1 ^ { 3 } = 132$ 
B)  $5 ^ { 3 } - 3 ^ { 3 } + 2 ^ { 3 } = 106$ 
C)  $5 ^ { 3 } + 3 ^ { 3 } - 2 ^ { 3 } + 1 ^ { 3 } = 145$ 
D)  $5 ^ { 3 } + 3 ^ { 3 } - 2 ^ { 3 } = 144$ 
A)  $5 ^ { 3 } + 2 ^ { 3 } - 1 ^ { 3 } = 132$ 
B)  $5 ^ { 3 } - 3 ^ { 3 } + 2 ^ { 3 } = 106$ 
C)  $5 ^ { 3 } + 3 ^ { 3 } - 2 ^ { 3 } + 1 ^ { 3 } = 145$ 
D)  $5 ^ { 3 } + 3 ^ { 3 } - 2 ^ { 3 } = 144$

Question 9

Use the formula indicated to determine the prime number generated for the given value of n.
-$p = n ^ { 2 } - n + 41 ; n = 34$

Accepted Answer

A) 1163 
B) 3801 
C) 2241 
D) 1149 
A) 1163 
B) 3801 
C) 2241 
D) 1149

Question 10

For the following amicable pair, determine whether neither, one, or both of the members are happy, and whether the pair is a happy amicable pair.
-32,685,250 and 34,538,270

Accepted Answer

A) one; happy 
B) both; happy 
C) one; not happy 
D) neither; not happy 
A) one; happy 
B) both; happy 
C) one; not happy 
D) neither; not happy

Question 11

A prime number may have one or two different natural number factors but may not have more
than two different natural number factors.

Accepted Answer

A) True 
 B)False

Question 12

Determine whether the statement is true or false.
-Every natural number of the form 4k + 4 is prime.

Accepted Answer

A) True 
 B)False

Question 13

Determine whether the statement is true or false.
-If you square any term of the Lucas sequence and subtract 5 you will obtain the product of the 2 terms either side of the term squared.

Accepted Answer

A) True 
 B)False

Question 14

Write the number as the sum of two primes. There may be more than one way to do this.
-34

Accepted Answer

A) 29 + 5, 31 + 3, 23 + 11, 17 + 17 
B) 15 + 19, 29 + 5, 23 + 11 
C) 21 + 13, 31 + 3, 29 + 5 
D) 27 + 7, 29 + 5, 23 + 11 
A) 29 + 5, 31 + 3, 23 + 11, 17 + 17 
B) 15 + 19, 29 + 5, 23 + 11 
C) 21 + 13, 31 + 3, 29 + 5 
D) 27 + 7, 29 + 5, 23 + 11

Question 15

Find the residue.
-86 (mod 8)

Accepted Answer

A) 5 
B) 8 
C) 6 
D) 7 
A) 5 
B) 8 
C) 6 
D) 7

Question 16

Find the number of divisors of the number.
-400

Accepted Answer

A) 16 
B) 8 
C) 12 
D) 15 
A) 16 
B) 8 
C) 12 
D) 15

Question 17

Find the greatest common factor of the numbers in the group.
-20, 30, 35, 40

Accepted Answer

A) 5 
B) 1 
C) 4 
D) 20 
A) 5 
B) 1 
C) 4 
D) 20

Question 18

Find the residue.
-6 (mod 20)

Accepted Answer

A) 18 
B) 15 
C) 16 
D) 17 
A) 18 
B) 15 
C) 16 
D) 17

Question 19

Solve the problem relating to the Fibonacci sequence.
-According to the Binet form, the nth Fibonacci number is given by $\frac { \left( \frac { 1 + \sqrt { 5 } } { 2 } \right) ^ { n } - \left( \frac { 1 - \sqrt { 5 } } { 2 } \right) ^ { n } } { \sqrt { 5 } }$ Use this Binet form to find the 28th Fibonacci number.

Accepted Answer

A) 317,812 
B) 317,810 
C) 317,811 
D) 317,813 
A) 317,812 
B) 317,810 
C) 317,811 
D) 317,813

Question 20

Determine whether the statement is true or false.
-The set of all common multiples of two given whole numbers is finite.

Accepted Answer

A) True 
 B)False

-=3 -=8 -=21

Find the number of divisors of the number. -60

1=2-1 1+1=3-1 1+1+2=5-1

Determine whether the statement is true or false. -If a > b > c, then the least common multiple of p^aq^b, p^bq^c, and p^cq^a is p^aq^a.

Find all natural number factors of the number. -42

Answer the question. -Planets A, B, and C orbit a certain star once every 3, 7, and 18 months, respectively. If the three planets are now in the same straight line, what is the smallest number of months that must pass Before they line up again?

Find all natural number factors of the number. -125

+-=8 +-=34

Use the formula indicated to determine the prime number generated for the given value of n. - $p = n ^ { 2 } - n + 41 ; n = 34$

For the following amicable pair, determine whether neither, one, or both of the members are happy, and whether the pair is a happy amicable pair. -32,685,250 and 34,538,270

A prime number may have one or two different natural number factors but may not have more than two different natural number factors.

Determine whether the statement is true or false. -Every natural number of the form 4k + 4 is prime.

Determine whether the statement is true or false. -If you square any term of the Lucas sequence and subtract 5 you will obtain the product of the 2 terms either side of the term squared.

Write the number as the sum of two primes. There may be more than one way to do this. -34

Find the residue. -86 (mod 8)

Find the number of divisors of the number. -400

Find the greatest common factor of the numbers in the group. -20, 30, 35, 40

Find the residue. -6 (mod 20)

Determine whether the statement is true or false. -The set of all common multiples of two given whole numbers is finite.

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

-=3 -=8 -=21

Find the number of divisors of the number. -60

1=2-1 1+1=3-1 1+1+2=5-1

Determine whether the statement is true or false. -If a > b > c, then the least common multiple of paqb, pbqc, and pcqa is paqa.

Find all natural number factors of the number. -42

Answer the question. -Planets A, B, and C orbit a certain star once every 3, 7, and 18 months, respectively. If the three planets are now in the same straight line, what is the smallest number of months that must pass Before they line up again?

Find all natural number factors of the number. -125

+-=8 +-=34

Use the formula indicated to determine the prime number generated for the given value of n. - p=n2−n+41;n=34p = n ^ { 2 } - n + 41 ; n = 34p=n2−n+41;n=34

For the following amicable pair, determine whether neither, one, or both of the members are happy, and whether the pair is a happy amicable pair. -32,685,250 and 34,538,270

A prime number may have one or two different natural number factors but may not have more than two different natural number factors.

Determine whether the statement is true or false. -Every natural number of the form 4k + 4 is prime.

Determine whether the statement is true or false. -If you square any term of the Lucas sequence and subtract 5 you will obtain the product of the 2 terms either side of the term squared.

Write the number as the sum of two primes. There may be more than one way to do this. -34

Find the residue. -86 (mod 8)

Find the number of divisors of the number. -400

Find the greatest common factor of the numbers in the group. -20, 30, 35, 40

Find the residue. -6 (mod 20)

Determine whether the statement is true or false. -The set of all common multiples of two given whole numbers is finite.

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

Determine whether the statement is true or false. -If a > b > c, then the least common multiple of p^aq^b, p^bq^c, and p^cq^a is p^aq^a.

Use the formula indicated to determine the prime number generated for the given value of n. - $p = n ^ { 2 } - n + 41 ; n = 34$