Question 1

Determine whether the statement is true or false.
-Every natural number can be expressed as a sum of Fibonacci numbers, where no number is used more than once.

Accepted Answer

A) True 
 B)False

Question 2

Determine whether the statement is true or false.
-Two natural, relatively prime numbers have at most one common factor.

Accepted Answer

A) True 
 B)False

Question 3

Give the prime factorization of the number. Use exponents when possible.
-396

Accepted Answer

A)  $3^4 \cdot 11$ 
B)  $2^4 \cdot 11$ 
C)  $2 ^ { 2 } \cdot 3 ^ { 2 } \cdot 11$ 
D)  $2 ^ { 3 } \cdot 3 ^ { 2 } \cdot 11$ 
A)  $3^4 \cdot 11$ 
B)  $2^4 \cdot 11$ 
C)  $2 ^ { 2 } \cdot 3 ^ { 2 } \cdot 11$ 
D)  $2 ^ { 3 } \cdot 3 ^ { 2 } \cdot 11$

Question 4

Determine whether the statement is true or false.
-Greatest common factors can not be found by using prime factorization, only by using the division method.

Accepted Answer

A) True 
 B)False

Question 5

All prime numbers are odd.

Accepted Answer

A) True 
 B)False

Question 6

$\begin{aligned}
1 ^ { 2 } & = 1 \cdot 1 \
1 ^ { 2 } + 1 ^ { 2 } & = 1 \cdot 2 \
1 ^ { 2 } + 1 ^ { 2 } + 2 ^ { 2 } & = 2 \cdot 3
\end{aligned}$

Accepted Answer

A)  $1 ^ { 2 } + 1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } = 3 \cdot 5$ 
B)  $1 ^ { 2 } + 1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + 5 ^ { 2 } = 5 \cdot 8$ 
C)  $1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } = 2 \cdot 7$ 
D)  $1 ^ { 2 } + 1 ^ { 2 } + 2 ^ { 2 } + 2 ^ { 2 } = 2 \cdot 5$ 
A)  $1 ^ { 2 } + 1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } = 3 \cdot 5$ 
B)  $1 ^ { 2 } + 1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + 5 ^ { 2 } = 5 \cdot 8$ 
C)  $1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } = 2 \cdot 7$ 
D)  $1 ^ { 2 } + 1 ^ { 2 } + 2 ^ { 2 } + 2 ^ { 2 } = 2 \cdot 5$

Question 7

Every composite number is divisible by 2.

Accepted Answer

A) True 
 B)False

Question 8

Give the prime factorization of the number. Use exponents when possible.
-198

Accepted Answer

A)  $2 \cdot 3 \cdot 11$ 
B)  $2 ^ { 2 } \cdot 3 ^ { 2 } \cdot 11$ 
C)  $22 \cdot 3 ^ { 2 }$ 
D)  $2 \cdot 3 ^ { 2 } \cdot 11$ 
A)  $2 \cdot 3 \cdot 11$ 
B)  $2 ^ { 2 } \cdot 3 ^ { 2 } \cdot 11$ 
C)  $22 \cdot 3 ^ { 2 }$ 
D)  $2 \cdot 3 ^ { 2 } \cdot 11$

Question 9

Write the number as the sum of two primes. There may be more than one way to do this.
-28

Accepted Answer

A) 5 + 23, 13 + 15 
B) 14 + 14 
C) 3 + 25, 5 + 23, 7 + 21 
D) 5 + 23, 11 + 17 
A) 5 + 23, 13 + 15 
B) 14 + 14 
C) 3 + 25, 5 + 23, 7 + 21 
D) 5 + 23, 11 + 17

Question 10

Find the number of divisors of the number.
-360

Accepted Answer

A) 36 
B) 12 
C) 24 
D) 18 
A) 36 
B) 12 
C) 24 
D) 18

Question 11

Answer the question.
-Two runners run around a circular track. The first runner completes each lap in 6 minutes. The second runner completes each lap in 13 minutes. If they both start at the same place and the same
Time and go in the same direction, after how many minutes will they meet again at the starting
Place?

Accepted Answer

A) 78 
B) 156 
C) 79 
D) 19 
A) 78 
B) 156 
C) 79 
D) 19

Question 12

Answer the question.
-Three taxi cabs make a complete trip from downtown to the airport and back in 10, 26 and 65 minutes, respectively. If all three cabs leave at the same time, what is the shortest time that must
Pass before they are all together again?

Accepted Answer

A) 101 minutes 
B) 260 minutes 
C) 1690 minutes 
D) 130 minutes 
A) 101 minutes 
B) 260 minutes 
C) 1690 minutes 
D) 130 minutes

Question 13

Determine whether the statement is true or false.
-$\text { If } 2 ^ { n } - 1 \text { is prime, then } 2 ^ { n - 1 } \left( 2 ^ { n } - 1 \right) \text { is perfect. }$

Accepted Answer

A) True 
 B)False

Question 14

Determine all values for the digit x that make the first number divisible by the second number. If none exist, so state.
-43x0 is divisible by 6.

Accepted Answer

The answer of Determine all values for the digit x...

Question 15

Determine whether the statement is true or false.
-$$\text { If } \mathrm { a } > \mathrm { b } > \mathrm { c } \text {, then the greatest common factor of } \mathrm { p } ^ { \mathrm { a } } \mathrm { q } ^ { \mathrm { b } } , \mathrm { p } ^ { \mathrm { b } } \mathrm { q } ^ { \mathrm { c } } \text {, and } \mathrm { p } ^ { \mathrm { c } ^ { \mathrm { a } } } \text { is } \mathrm { p } ^ { \mathrm { a } } \mathrm { q } ^ { \mathrm { a } }$$

Accepted Answer

A) True 
 B)False

Question 16

If a natural number is divisible by 3 and 5, then it must also be divisible by 15.

Accepted Answer

A) True 
 B)False

Question 17

Find the least common multiple of the numbers in the group.
-30, 40, 70

Accepted Answer

A) 280 
B) 120 
C) 420 
D) 840 
A) 280 
B) 120 
C) 420 
D) 840

Question 18

Determine all values for the digit x that make the first number divisible by the second number. If none exist, so state.
-74,3x2 is divisible by 6.

Accepted Answer

The answer of Determine all values for the digit x...

Question 19

Determine whether the statement is true or false.
-The following quotient, where Fn represents the nth term of the Fibonacci sequence, approaches the golden ratio as n gets larger: $\frac { \mathrm { F } _ { \mathrm { n } } + 1 } { \mathrm {~F} _ { \mathrm { n } } }$

Accepted Answer

A) True 
 B)False

Question 20

Find the least common multiple of the numbers in the group.
-48, 162, 27

Accepted Answer

A) 1296 
B) 432 
C) 648 
D) 324 
A) 1296 
B) 432 
C) 648 
D) 324

Determine whether the statement is true or false. -Every natural number can be expressed as a sum of Fibonacci numbers, where no number is used more than once.

Determine whether the statement is true or false. -Two natural, relatively prime numbers have at most one common factor.

Give the prime factorization of the number. Use exponents when possible. -396

Determine whether the statement is true or false. -Greatest common factors can not be found by using prime factorization, only by using the division method.

All prime numbers are odd.

=1\cdot1 + =1\cdot2 ++ =2\cdot3

Every composite number is divisible by 2.

Give the prime factorization of the number. Use exponents when possible. -198

Write the number as the sum of two primes. There may be more than one way to do this. -28

Find the number of divisors of the number. -360

Answer the question. -Three taxi cabs make a complete trip from downtown to the airport and back in 10, 26 and 65 minutes, respectively. If all three cabs leave at the same time, what is the shortest time that must Pass before they are all together again?

Determine whether the statement is true or false. - $\text { If } 2 ^ { n } - 1 \text { is prime, then } 2 ^ { n - 1 } \left( 2 ^ { n } - 1 \right) \text { is perfect. }$

Determine all values for the digit x that make the first number divisible by the second number. If none exist, so state. -43x0 is divisible by 6.

If a natural number is divisible by 3 and 5, then it must also be divisible by 15.

Find the least common multiple of the numbers in the group. -30, 40, 70

Determine all values for the digit x that make the first number divisible by the second number. If none exist, so state. -74,3x2 is divisible by 6.

Determine whether the statement is true or false. -The following quotient, where Fn represents the nth term of the Fibonacci sequence, approaches the golden ratio as n gets larger: $\frac { \mathrm { F } _ { \mathrm { n } } + 1 } { \mathrm {~F} _ { \mathrm { n } } }$

Find the least common multiple of the numbers in the group. -48, 162, 27

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

Determine whether the statement is true or false. -Every natural number can be expressed as a sum of Fibonacci numbers, where no number is used more than once.

Determine whether the statement is true or false. -Two natural, relatively prime numbers have at most one common factor.

Give the prime factorization of the number. Use exponents when possible. -396

Determine whether the statement is true or false. -Greatest common factors can not be found by using prime factorization, only by using the division method.

All prime numbers are odd.

=1\cdot1 + =1\cdot2 ++ =2\cdot3

Every composite number is divisible by 2.

Give the prime factorization of the number. Use exponents when possible. -198

Write the number as the sum of two primes. There may be more than one way to do this. -28

Find the number of divisors of the number. -360

Answer the question. -Three taxi cabs make a complete trip from downtown to the airport and back in 10, 26 and 65 minutes, respectively. If all three cabs leave at the same time, what is the shortest time that must Pass before they are all together again?

Determine whether the statement is true or false. - If 2n−1 is prime, then 2n−1(2n−1) is perfect. \text { If } 2 ^ { n } - 1 \text { is prime, then } 2 ^ { n - 1 } \left( 2 ^ { n } - 1 \right) \text { is perfect. } If 2n−1 is prime, then 2n−1(2n−1) is perfect.

Determine all values for the digit x that make the first number divisible by the second number. If none exist, so state. -43x0 is divisible by 6.

If a natural number is divisible by 3 and 5, then it must also be divisible by 15.

Find the least common multiple of the numbers in the group. -30, 40, 70

Determine all values for the digit x that make the first number divisible by the second number. If none exist, so state. -74,3x2 is divisible by 6.

Determine whether the statement is true or false. -The following quotient, where Fn represents the nth term of the Fibonacci sequence, approaches the golden ratio as n gets larger: Fn+1 Fn\frac { \mathrm { F } _ { \mathrm { n } } + 1 } { \mathrm {~F} _ { \mathrm { n } } } Fn​Fn​+1​

Find the least common multiple of the numbers in the group. -48, 162, 27

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. - $\text { If } 2 ^ { n } - 1 \text { is prime, then } 2 ^ { n - 1 } \left( 2 ^ { n } - 1 \right) \text { is perfect. }$

Determine whether the statement is true or false. -The following quotient, where Fn represents the nth term of the Fibonacci sequence, approaches the golden ratio as n gets larger: $\frac { \mathrm { F } _ { \mathrm { n } } + 1 } { \mathrm {~F} _ { \mathrm { n } } }$