Number Theory

Problem 1

Find the opposite of the integer $-4$ .

See Solution

Problem 2

Express each number as a power: $36 = 6^2$ , $49 = 7^2$ , $81 = 3^4$ , $100 = 10^2$ .

See Solution

Problem 3

Find the prime factorization of 25 and 42.

See Solution

Problem 4

Which isotope of oxygen is not naturally occurring? $\{$ a. Oxygen-16, b. Oxygen-17, c. Ozone, d. Oxygen-15 $\}$

See Solution

Problem 5

Find the error interval for a number $x$ rounded to 1 significant figure as $0.08$ .

See Solution

Problem 6

Find the last digit of $33^{333}$ .

See Solution

Problem 7

Compare the two 8s in the number 8,825.

See Solution

Problem 8

Determine if each rational number $\frac{a}{b}$ is a perfect square.

See Solution

Problem 9

Find all integer solutions to the linear equation $3x + 7y = 2$ .

See Solution

Problem 10

Determine the base of $-(-5)^{3}$ and evaluate $6^{5}$ .

See Solution

Problem 11

Determine if the statement "Any prime number must be deficient" is true or false.

See Solution

Problem 12

Express the set of natural numbers between 3 and 80 in roster form: $\{x \mid x \in \mathbb{N}, 3 \leq x < 80\}$ .

See Solution

Problem 13

Convert decimal number $634_{10}$ to its octal representation.

See Solution

Problem 14

Find the prime factorization of $112$ .

See Solution

Problem 15

Find the prime factorization of 90 using exponents.

See Solution

Problem 16

Find the largest value of $n$ such that $5^{n}$ divides $1 \times 2 \times 3 \times \cdots \times 15$ .

See Solution

Problem 17

Prove the statement "if $2^{x} = 3$ , then $x$ is not rational" is logically equivalent to the original statement "if $x$ is rational, then $2^{x} \neq 3$ ".

See Solution

Problem 18

What is the number of real fifth roots of 0?

See Solution

Problem 19

Find the least common denominator (LCD) of 25, 50, and $75$ .

See Solution

Problem 20

Find the smallest integer $N$ such that the harmonic series $\sum_{k=1}^{N} \frac{1}{k}$ exceeds 4.

See Solution

Problem 21

Prove that there are infinitely many primes of the form $3k+2$ , where $k$ is a natural number.

See Solution

Problem 22

Find the integer between 9 and 999 with the most positive divisors.

See Solution

Problem 23

Find the integer $x$ such that $-84 \equiv x \pmod{15}$ .

See Solution

Problem 24

Convert the binary number 110011 to its decimal equivalent.

See Solution

Problem 25

Find the two integers that bound the square root of 115.

See Solution

Problem 26

Determine which of the following are integers: -95, $\frac{9}{10}$ , $-1 \frac{26}{43}$ , -36.

See Solution

Problem 27

Determine which statements are true: $\sqrt{4}$ is rational and integer, $\sqrt{3}$ is rational, 0 is neither rational nor irrational, $-6.1\overline{33}$ is irrational.

See Solution

Problem 28

Find an integer that has remainders of 2, 2, and 3 when divided by 3, 7, and 5 respectively, as described by Sun Zi's Chinese Remainder Theorem.

See Solution

Problem 29

Find the true statement about the opposite and absolute value of 45 and -45.
The opposite of -45 is equal to the absolute value of -45.

See Solution

Problem 30

Geben Sie Zahlen an, die: a) ganze, aber keine rationalen Zahlen sind. $/(v) ?$ b) Brüche und negative Zahlen sind. $-\frac{2 y}{4},-\frac{1}{2}$ c) reell, aber nicht rational sind. d) irrationale Zahlen sind, deren Quadrat natürliche Zahlen sind.

See Solution

Problem 31

Choose the correct integers $y \geq -1$ for the set $\{y: y \text{ is an integer and } y \geq -1\}$ . Choose 1 option, 5 points.

See Solution

Problem 32

Find the numbers with absolute values of $3$ , $\frac{5}{3}$ , and $9 \frac{1}{2}$ .

See Solution

Problem 33

Find the value of $2^{-5}$ .

See Solution

Problem 34

Find the value of $\left(10^{5}\right)^{6}$ . Options: $10^{30}$ , $10^{11}$ , $10^{3}$ .

See Solution

Problem 35

Determine if the number $97$ is divisible by $2, 3, 4, 5, 6, 8, 9$ , and/or $10$ . If not, answer "None".

See Solution

Problem 36

Find the minimum number of prizes that can be distributed from $\$800$ , where the prizes are powers of 2.

See Solution

Problem 37

Find a real number $x$ such that $x > 0.48$ .

See Solution

Problem 38

Find the next whole number after $A18A_{11}$ in base-11 number system.

See Solution

Problem 39

Explain why $3^{0}$ and $7^{0}$ are both equal to 1.

See Solution

Problem 40

Find the prime factorization of 125. $5 \times 5 \times 5$ , $5 \times 25$ , or $5 \times 100$ .

See Solution

Problem 41

Rewrite scientific notation problems: $27, 607, 3507, 0.01, 0.0085, 8.1\times 10^{-6}$ , $419, 4126, 7053, 0.68, 4.9\times 10^{-3}, 8.07\times 10^{-9}$ , $810, 8540, 8.56\times 10^{5}, 4.8\times 10^{-3}, 7.3\times 10^{-5}, 4.82\times 10^{8}$ .

See Solution

Problem 42

Find the prime factorization of 72 using exponential notation: $2^{3} \cdot 3^{2}$

See Solution

Problem 43

Simplify the expression $\frac{1}{256^{101}}$ as a whole number with a negative exponent.

See Solution

Problem 44

Convert 823 to base-5 notation. $823=\square_{\text{five}}$

See Solution

Problem 45

Find the digit in the hundred thousands place of the cost to produce a movie that cost $\$ 3,254,107$ .

See Solution

Problem 46

1. Order $-42, -13, 6, 38$ from least to greatest: $-42 \leq -13 \leq 6 \leq 38$ . 2. Order $-6 \frac{1}{2}, -5 \frac{1}{2}, -4, -6$ from least to greatest: $-6 \frac{1}{2} \leq -6 \leq -5 \frac{1}{2} \leq -4$ . 3. Order $-8.999, 0, 17.56, -8 \frac{2}{3}$ from least to greatest: $-8.999 \leq 0 \leq -8 \frac{2}{3} \leq 17.56$ . 4. Order $-\frac{4}{10}, \frac{1}{3}, 1 \frac{8}{9}, -5$ from least to greatest: $-5 \leq -\frac{4}{10} \leq \frac{1}{3} \leq 1 \frac{8}{9}$ .

See Solution

Problem 47

Find the place value of the digit 4 in the number $814,592$ .

See Solution

Problem 48

What does $5.2 \mathrm{E}-7$ represent on a calculator? (0.000000052)

See Solution

Problem 49

Convert measurements to scientific notation: a. $1.2 \times 10^9 \mathrm{dL}$ b. $6.7 \times 10^{-11} \mathrm{mm}$ c. $6.7 \times 10^{10} \mathrm{g}$

See Solution

Problem 50

Choose all true statements about the imaginary unit $i$ : $i^{2}=-1$ , $i=\sqrt{-1}$ , the solution of $x^{2}=-1$ .

See Solution

Problem 51

Determine if the solution to the equation $B=0.181818181818\dots$ is a rational or irrational number.

See Solution

Problem 52

Evaluate the expression $2^{10}$ .

See Solution

Problem 53

Find the percent abundance of Cu-63 (62.9296 u) and Cu-65 (64.9278 u) given copper's atomic weight of $63.546 u$ .

See Solution

Problem 54

Complete a table to determine if each number is a factor of 12, a multiple of 12, or both. Numbers: $12$ , $24$ , $36$ .

See Solution

Problem 55

Is $-\sqrt{0/4}$ a rational number?

See Solution

Problem 56

Find the prime factorization of 124.

See Solution

Problem 57

Identify all composite numbers among 44, 28, and 37.

See Solution

Problem 58

Simplify the complex number $i^{-28}$ to its most basic form.

See Solution

Problem 59

Evaluate the expression $9! / 4!$ , where $n!$ represents the factorial of $n$ .

See Solution

Problem 60

Solve the modular linear equation $19x \equiv 4 \pmod{141}$ for $x$ .

See Solution

Problem 61

Find the number of powers with a multiple of 4 first. Simplify $i^{10}$ .

See Solution

Problem 62

Find the greatest common factor (GCF) of $75$ and $350$ .

See Solution

Problem 63

Convert the given base 4 number $12131_{\text{four}}$ to base 10.

See Solution

Problem 64

The set of all real numbers $x \geq -1.8$ .

See Solution

Problem 65

Find the number of 5-digit numbers whose digit product is $6!$ .

See Solution

Problem 66

Find the greatest number of identical flower arrangements Lauren can make using $49$ roses and $28$ daisies, with no flowers left over.

See Solution

Problem 67

Amelia thinks of an odd number that is a factor of both $18$ and $12$ . What is the largest number she could be thinking of?

See Solution

Problem 68

Find a real number $x$ such that $-7.81 < x < -7$ .

See Solution

Problem 69

Emma's number is between 825 and 875, inclusive. Find the smallest and largest possible values.

See Solution

Problem 70

Find the prime factor tree of 330 and use it to calculate the highest common factor (HCF) of 308 and 330.

See Solution

Problem 71

Determine if $0$ is a perfect square and justify your answer.

See Solution

Problem 72

Find the least residue of $50 \pmod{3}, 50 \pmod{4}, 55 \pmod{7}, 34 \pmod{4}, 75 \pmod{5}$ and show your work.

See Solution

Problem 73

Find the smallest product of two numbers from the given values in standard form: $4.5 \times 10^{5}$ , $1 \times 10^{6}$ , $8 \times 10^{3}$ , $5 \times 10^{5}$ .

See Solution

Problem 74

Simplify $b^{0}$ where $b \neq 0$ . Simplify the result.

See Solution

Problem 75

Rewrite the set V by listing its elements using appropriate set notation. $V=\{y | -4 < y < -1, y \in \mathbb{Z}\}$

See Solution

Problem 76

Student claims 84914 > 311902 because 8 > 3. Determine if the student's reasoning is correct.

See Solution

Problem 77

Amelia tried to write 2400 in standard index form. Explain her mistake and provide the correct answer: $2400=2.4 \times 10^3$ .

See Solution

Problem 78

Solve the exponential equation $7^{x}=1$ for the value of $x$ .

See Solution

Problem 79

Classify numbers as rational or irrational: $-28.\overline{17}$ , $-\sqrt{16}$ , $\frac{6}{19}$ , $5\pi$ , $\sqrt{22}$ .

See Solution

Problem 80

Plot the set $\{-3.5, -2.5, -2.0, 1.5\}$ on the number line.

See Solution

Problem 81

Determine if the set $\{x \mid x \text{ is a real number larger than 18}\}$ is finite or infinite.

See Solution

Problem 82

Find the least common multiple (LCM) of $3, 8, 12$ .

See Solution

Problem 83

Find the greatest common factor of $16$ and $40$ . The greatest common factor is $\square$ .

See Solution

Problem 84

Determine if $0.393993999399993999993\ldots$ is rational or irrational. The pattern of digits repeating indefinitely indicates the number is rational.

See Solution

Problem 85

Find the value of each digit in the number $8,236,014,597$ .

See Solution

Problem 86

Find two numbers whose lowest common multiple is 90 and highest common factor is 6.

See Solution

Problem 87

Find the number whose factors include $1, 2, 3, 4, 6$ , and $12$ .

See Solution

Problem 88

Find the Pythagorean triple among the given sets of numbers: $\{19,121,122\}$ , $\{27,363,364\}$ , $\{35,612,613\}$ , $\{8,15,16\}$ .

See Solution

Problem 89

Convert base 8 number 4,305 to hexadecimal.
$4,305_{8} = \square_{16}$

See Solution

Problem 90

Find the square root of 784 using prime factorization.

See Solution

Problem 91

Find the number of unique passwords with 2 digits, where each digit can be a number or a letter.
Solution: There are 10 digits (0-9) and 26 letters (A-Z), so each digit can be chosen from 36 possibilities. Therefore, the total number of possible passwords is $36 \times 36 = 1,296$ .

See Solution

Problem 92

Find the value of $x$ in the equation $x=\sqrt[3]{(-245)^{-\frac{1}{5}}}$ rounded to the nearest hundredth.

See Solution

Problem 93

Trouve l'ensemble de nombres quantiques non permis et modifie un nombre pour obtenir un ensemble permis. $n=1, l=2, m_{l}=-2$ ou $n=4, l=1, m_{l}=-2$

See Solution

Problem 94

Find the set of counting numbers less than or equal to 3, expressed using set notation and listing method. $\{1, 2, 3\}$

See Solution

Problem 95

Find the base $\mathrm{x}$ where $135_{10}=343_\mathrm{x}$

See Solution

Problem 96

Find the least possible number that when divided by 6 leaves a remainder of 2, and when divided by 7 leaves a remainder of 3. Determine if Daniel's claim that the least number is 38 is correct.

See Solution

Problem 97

Find the approximate value of the expression $2007! / 2005!$ .

See Solution

Problem 98

Ordenar $-3, 0, -8, -9$ de menor a mayor.

See Solution

Problem 99

Calculate $\log_6 9$ using the change of base formula, rounding the answer to the nearest thousandth.

See Solution

Problem 100

Rewrite the calculator display $3.35 \mathrm{E}-5$ in scientific notation.

See Solution

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Number Theory

Problem 1

Find the opposite of the integer −4-4−4.

Problem 2

Express each number as a power: 36=6236 = 6^236=62, 49=7249 = 7^249=72, 81=3481 = 3^481=34, 100=102100 = 10^2100=102.

Problem 3

Find the prime factorization of 25 and 42.

Problem 4

Which isotope of oxygen is not naturally occurring? {\{{a. Oxygen-16, b. Oxygen-17, c. Ozone, d. Oxygen-15}\}}

Problem 5

Find the error interval for a number xxx rounded to 1 significant figure as 0.080.080.08.

Problem 6

Find the last digit of 3333333^{333}33333.

Problem 7

Compare the two 8s in the number 8,825.

Problem 8

Determine if each rational number ab\frac{a}{b}ba​ is a perfect square.

Problem 9

Find all integer solutions to the linear equation 3x+7y=23x + 7y = 23x+7y=2.

Problem 10

Determine the base of −(−5)3-(-5)^{3}−(−5)3 and evaluate 656^{5}65.

Problem 11

Determine if the statement "Any prime number must be deficient" is true or false.

Problem 12

Express the set of natural numbers between 3 and 80 in roster form: {x∣x∈N,3≤x<80}\{x \mid x \in \mathbb{N}, 3 \leq x < 80\}{x∣x∈N,3≤x<80}.

Problem 13

Convert decimal number 63410634_{10}63410​ to its octal representation.

Problem 14

Find the prime factorization of 112112112.

Problem 15

Find the prime factorization of 90 using exponents.

Problem 16

Find the largest value of nnn such that 5n5^{n}5n divides 1×2×3×⋯×151 \times 2 \times 3 \times \cdots \times 151×2×3×⋯×15.

Problem 17

Prove the statement "if 2x=32^{x} = 32x=3, then xxx is not rational" is logically equivalent to the original statement "if xxx is rational, then 2x≠32^{x} \neq 32x=3".

Problem 18

What is the number of real fifth roots of 0?

Problem 19

Find the least common denominator (LCD) of 25, 50, and 757575.

Problem 20

Find the smallest integer NNN such that the harmonic series ∑k=1N1k\sum_{k=1}^{N} \frac{1}{k}∑k=1N​k1​ exceeds 4.

Problem 21

Prove that there are infinitely many primes of the form 3k+23k+23k+2, where kkk is a natural number.

Problem 22

Find the integer between 9 and 999 with the most positive divisors.

Problem 23

Find the integer xxx such that −84≡x(mod15)-84 \equiv x \pmod{15}−84≡x(mod15).

Problem 24

Convert the binary number 110011 to its decimal equivalent.

Problem 25

Find the two integers that bound the square root of 115.

Problem 26

Determine which of the following are integers: -95, 910\frac{9}{10}109​, −12643-1 \frac{26}{43}−14326​, -36.

Problem 27

Determine which statements are true: 4\sqrt{4}4​ is rational and integer, 3\sqrt{3}3​ is rational, 0 is neither rational nor irrational, −6.133‾-6.1\overline{33}−6.133 is irrational.

Problem 28

Find an integer that has remainders of 2, 2, and 3 when divided by 3, 7, and 5 respectively, as described by Sun Zi's Chinese Remainder Theorem.

Problem 29

Find the true statement about the opposite and absolute value of 45 and -45.The opposite of -45 is equal to the absolute value of -45.

Problem 30

Geben Sie Zahlen an, die: a) ganze, aber keine rationalen Zahlen sind. /(v)?/(v) ?/(v)? b) Brüche und negative Zahlen sind. −2y4,−12-\frac{2 y}{4},-\frac{1}{2}−42y​,−21​ c) reell, aber nicht rational sind. d) irrationale Zahlen sind, deren Quadrat natürliche Zahlen sind.

Problem 31

Choose the correct integers y≥−1y \geq -1y≥−1 for the set {y:y is an integer and y≥−1}\{y: y \text{ is an integer and } y \geq -1\}{y:y is an integer and y≥−1}. Choose 1 option, 5 points.

Problem 32

Find the numbers with absolute values of 333, 53\frac{5}{3}35​, and 9129 \frac{1}{2}921​.

Problem 33

Find the value of 2−52^{-5}2−5.

Problem 34

Find the value of (105)6\left(10^{5}\right)^{6}(105)6. Options: 103010^{30}1030, 101110^{11}1011, 10310^{3}103.

Problem 35

Determine if the number 979797 is divisible by 2,3,4,5,6,8,92, 3, 4, 5, 6, 8, 92,3,4,5,6,8,9, and/or 101010. If not, answer "None".

Problem 36

Find the minimum number of prizes that can be distributed from $800\$800$800, where the prizes are powers of 2.

Problem 37

Find a real number xxx such that x>0.48x > 0.48x>0.48.

Problem 38

Find the next whole number after A18A11A18A_{11}A18A11​ in base-11 number system.

Problem 39

Explain why 303^{0}30 and 707^{0}70 are both equal to 1.

Problem 40

Find the opposite of the integer $-4$ .

Express each number as a power: $36 = 6^2$ , $49 = 7^2$ , $81 = 3^4$ , $100 = 10^2$ .

Which isotope of oxygen is not naturally occurring? $\{$ a. Oxygen-16, b. Oxygen-17, c. Ozone, d. Oxygen-15 $\}$

Find the error interval for a number $x$ rounded to 1 significant figure as $0.08$ .

Find the last digit of $33^{333}$ .

Determine if each rational number $\frac{a}{b}$ is a perfect square.

Find all integer solutions to the linear equation $3x + 7y = 2$ .

Determine the base of $-(-5)^{3}$ and evaluate $6^{5}$ .

Express the set of natural numbers between 3 and 80 in roster form: $\{x \mid x \in \mathbb{N}, 3 \leq x < 80\}$ .

Convert decimal number $634_{10}$ to its octal representation.

Find the prime factorization of $112$ .

Find the largest value of $n$ such that $5^{n}$ divides $1 \times 2 \times 3 \times \cdots \times 15$ .

Prove the statement "if $2^{x} = 3$ , then $x$ is not rational" is logically equivalent to the original statement "if $x$ is rational, then $2^{x} \neq 3$ ".

Find the least common denominator (LCD) of 25, 50, and $75$ .

Find the smallest integer $N$ such that the harmonic series $\sum_{k=1}^{N} \frac{1}{k}$ exceeds 4.

Prove that there are infinitely many primes of the form $3k+2$ , where $k$ is a natural number.

Find the integer $x$ such that $-84 \equiv x \pmod{15}$ .

Determine which of the following are integers: -95, $\frac{9}{10}$ , $-1 \frac{26}{43}$ , -36.

Determine which statements are true: $\sqrt{4}$ is rational and integer, $\sqrt{3}$ is rational, 0 is neither rational nor irrational, $-6.1\overline{33}$ is irrational.

Find the true statement about the opposite and absolute value of 45 and -45.
The opposite of -45 is equal to the absolute value of -45.

Geben Sie Zahlen an, die: a) ganze, aber keine rationalen Zahlen sind. $/(v) ?$ b) Brüche und negative Zahlen sind. $-\frac{2 y}{4},-\frac{1}{2}$ c) reell, aber nicht rational sind. d) irrationale Zahlen sind, deren Quadrat natürliche Zahlen sind.

Choose the correct integers $y \geq -1$ for the set $\{y: y \text{ is an integer and } y \geq -1\}$ . Choose 1 option, 5 points.

Find the numbers with absolute values of $3$ , $\frac{5}{3}$ , and $9 \frac{1}{2}$ .

Find the value of $2^{-5}$ .

Find the value of $\left(10^{5}\right)^{6}$ . Options: $10^{30}$ , $10^{11}$ , $10^{3}$ .

Determine if the number $97$ is divisible by $2, 3, 4, 5, 6, 8, 9$ , and/or $10$ . If not, answer "None".

Find the minimum number of prizes that can be distributed from $\$800$ , where the prizes are powers of 2.

Find a real number $x$ such that $x > 0.48$ .

Find the next whole number after $A18A_{11}$ in base-11 number system.

Explain why $3^{0}$ and $7^{0}$ are both equal to 1.

Find the prime factorization of 125. $5 \times 5 \times 5$ , $5 \times 25$ , or $5 \times 100$ .

Find the prime factorization of 72 using exponential notation: $2^{3} \cdot 3^{2}$

Simplify the expression $\frac{1}{256^{101}}$ as a whole number with a negative exponent.

Convert 823 to base-5 notation. $823=\square_{\text{five}}$

Find the digit in the hundred thousands place of the cost to produce a movie that cost $\$ 3,254,107$ .

Find the place value of the digit 4 in the number $814,592$ .

What does $5.2 \mathrm{E}-7$ represent on a calculator? (0.000000052)

Convert measurements to scientific notation: a. $1.2 \times 10^9 \mathrm{dL}$ b. $6.7 \times 10^{-11} \mathrm{mm}$ c. $6.7 \times 10^{10} \mathrm{g}$

Choose all true statements about the imaginary unit $i$ : $i^{2}=-1$ , $i=\sqrt{-1}$ , the solution of $x^{2}=-1$ .

Determine if the solution to the equation $B=0.181818181818\dots$ is a rational or irrational number.

Evaluate the expression $2^{10}$ .

Find the percent abundance of Cu-63 (62.9296 u) and Cu-65 (64.9278 u) given copper's atomic weight of $63.546 u$ .

Complete a table to determine if each number is a factor of 12, a multiple of 12, or both. Numbers: $12$ , $24$ , $36$ .

Is $-\sqrt{0/4}$ a rational number?

Simplify the complex number $i^{-28}$ to its most basic form.

Evaluate the expression $9! / 4!$ , where $n!$ represents the factorial of $n$ .

Solve the modular linear equation $19x \equiv 4 \pmod{141}$ for $x$ .

Find the number of powers with a multiple of 4 first. Simplify $i^{10}$ .

Find the greatest common factor (GCF) of $75$ and $350$ .

Convert the given base 4 number $12131_{\text{four}}$ to base 10.

The set of all real numbers $x \geq -1.8$ .

Find the number of 5-digit numbers whose digit product is $6!$ .

Find the greatest number of identical flower arrangements Lauren can make using $49$ roses and $28$ daisies, with no flowers left over.

Amelia thinks of an odd number that is a factor of both $18$ and $12$ . What is the largest number she could be thinking of?

Find a real number $x$ such that $-7.81 < x < -7$ .

Determine if $0$ is a perfect square and justify your answer.

Find the least residue of $50 \pmod{3}, 50 \pmod{4}, 55 \pmod{7}, 34 \pmod{4}, 75 \pmod{5}$ and show your work.

Find the smallest product of two numbers from the given values in standard form: $4.5 \times 10^{5}$ , $1 \times 10^{6}$ , $8 \times 10^{3}$ , $5 \times 10^{5}$ .

Simplify $b^{0}$ where $b \neq 0$ . Simplify the result.