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by Matthew_07.
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Questions | Choices |

1. This number is both a Fibonacci number and a perfect cube. It is also the value of the number of vertices of any regular hexahedron. |
6 |

2. This Fibonacci number is the only prime that can be expressed as the sum of two consecutive prime numbers. This number is also the value of the length of the hypotenuse of the smallest possible right triangle with integer sides. |
8 |

3. This number is the smallest composite number. Any integer that is a multiple of this number can be expressed as a difference of two square numbers. |
7 |

4. This square number is an exponential factorial. Any number whose repeated digital sum equals this number is divisible by this number. |
3 |

5. This number, with a value identical to its square, square root, cube, and cube root, is the multiplicative identity of real numbers. |
2 |

6. This happy prime number is also the number of mathematical problems that are included in the Millennium Prize Problems. |
1 |

7. This composite number is the smallest perfect number. It is also the only number that can be expressed as the sum and the product of three consecutive positive integers. |
4 |

8. This number is the used as the base of the commonly-used decimal numeral system. It is also the sum of the first four factorials. |
5 |

9. This prime number is used as a rough estimate of two famous mathematical constants - pi and Euler's number. |
9 |

10. This prime number is associated with the mathematical concept of parity. The square root of this number is irrational. |
10 |

Quiz Answer Key and Fun Facts

Answer:
**8**

The two Fibonacci numbers that are also perfect cubes are 1 and 8. Note that 8 = 2*2*2 = 2^3.

A regular hexahedron, or simply, a cube, has six square faces, eight vertices, and 12 edges.

The two Fibonacci numbers that are also perfect cubes are 1 and 8. Note that 8 = 2*2*2 = 2^3.

A regular hexahedron, or simply, a cube, has six square faces, eight vertices, and 12 edges.

Answer:
**5**

The first few Fibonacci numbers are 1, 2, 3, 5, 8, and 13. Observe that 5 = 2 + 3, where both 2 and 3 are prime numbers.

A well-known primitive Pythagorean triple is given by (3,4,5), where 3^2 + 4^2 = 5^2, with 5 being the longest side or the hypotenuse.

The first few Fibonacci numbers are 1, 2, 3, 5, 8, and 13. Observe that 5 = 2 + 3, where both 2 and 3 are prime numbers.

A well-known primitive Pythagorean triple is given by (3,4,5), where 3^2 + 4^2 = 5^2, with 5 being the longest side or the hypotenuse.

Answer:
**4**

A composite number can be expressed as a product of smaller prime numbers. The number 4 is a composite number because 4 = 2 x 2. In fact, 4 is the smallest composite number, followed by 6. Note that 6 = 2 x 3.

Observe the equation 4*n = (n+1)^2 - (n-1)^2. This proves that for any integer which is a multiple of 4, it can be expressed as a difference of the two square numbers, n+1 and n-1. For example, when n = 2, 4*2 = 8 = 3^2 - 1^2.

A composite number can be expressed as a product of smaller prime numbers. The number 4 is a composite number because 4 = 2 x 2. In fact, 4 is the smallest composite number, followed by 6. Note that 6 = 2 x 3.

Observe the equation 4*n = (n+1)^2 - (n-1)^2. This proves that for any integer which is a multiple of 4, it can be expressed as a difference of the two square numbers, n+1 and n-1. For example, when n = 2, 4*2 = 8 = 3^2 - 1^2.

Answer:
**9**

The number 9 is a square number. Note that 9 = 3*3 = 3^2.

The number 9 can also be expressed in the form of 9 = 3^(2^1). As such, 9 is an exponential factorial. Another example of an exponential factorial is 262,144, where 262,144 = 4^(3^(2^1)).

The digital root, or the repeated sum digit of any number is computed by summing up the individual digits repeatedly, until a single value is obtained. For example, the digital root of the number 123 is 1 + 2 + 3 = 6. For the number 2468, we first perform the addition operation 2+4+6+8 to obtain 20. Subsequently, 2 + 0 = 2. Hence, the digital root of the number 2468 is 2.

The digital root of the number 27 is 2 + 7 = 9, implying that 27 is divisible by 9. Another example would be the number 12,345,678, where first, we obtain 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36, and subsequently 3 + 6 = 9, which implies that the number 12,345,678 is divisible by 9.

The number 9 is a square number. Note that 9 = 3*3 = 3^2.

The number 9 can also be expressed in the form of 9 = 3^(2^1). As such, 9 is an exponential factorial. Another example of an exponential factorial is 262,144, where 262,144 = 4^(3^(2^1)).

The digital root, or the repeated sum digit of any number is computed by summing up the individual digits repeatedly, until a single value is obtained. For example, the digital root of the number 123 is 1 + 2 + 3 = 6. For the number 2468, we first perform the addition operation 2+4+6+8 to obtain 20. Subsequently, 2 + 0 = 2. Hence, the digital root of the number 2468 is 2.

The digital root of the number 27 is 2 + 7 = 9, implying that 27 is divisible by 9. Another example would be the number 12,345,678, where first, we obtain 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36, and subsequently 3 + 6 = 9, which implies that the number 12,345,678 is divisible by 9.

Answer:
**1**

The number 1 is the first positive integer. Its square (1^2 =1*1 = 1), square root, cube (1^3 = 1*1*1 = 1), and cube root are all 1.

In addition, 1, being the multiplicative identity of real numbers, implies that 1*a = a, where a is any real number. For example, if a = 5, then 1*5 = 5. In other words, any real number, when multiplied with 1, will yield the same real number that we begin with.

On the other hand, 0 is the addition identity of real numbers. This implies 0 + b = b, where b is any real number. To illustrate, say we choose b to be 2. We note that 0 + 2 = 2.

The number 1 is the first positive integer. Its square (1^2 =1*1 = 1), square root, cube (1^3 = 1*1*1 = 1), and cube root are all 1.

In addition, 1, being the multiplicative identity of real numbers, implies that 1*a = a, where a is any real number. For example, if a = 5, then 1*5 = 5. In other words, any real number, when multiplied with 1, will yield the same real number that we begin with.

On the other hand, 0 is the addition identity of real numbers. This implies 0 + b = b, where b is any real number. To illustrate, say we choose b to be 2. We note that 0 + 2 = 2.

Answer:
**7**

The number 7 is considered to be a lucky number in both Western and Chinese cultures.

The number 7 is a prime number. A happy number is a number that yields a final value of 1, when the individual digits of that number are repeatedly squared and added up. Starting with 7, we have 7^2= 49, and subsequently 4^2 + 9^2 = 16 + 81 = 97. Then, 9^2 + 7^2 = 81 + 49 = 130. Next, 1^2 + 3^2 + 0^2 = 1 + 9 = 10. Finally, 1^2 + 0^2 = 1.

The Millennium Prize Problems consist of seven challenging mathematical problems. The list was compiled by the Clay Mathematics Institute in the year 2000. One of the problems, the Poincaré conjecture, was solved by Russian mathematician Grigori Perelman in 2003, but he declined the USD 1 million prize money.

The remaining six problems are the Birch and Swinnerton-Dyer conjecture, the Hodge conjecture, the Navier-Stokes existence and smoothness problem, the P versus NP problem, the Riemann hypothesis, and the Yang-Mills existence and mass gap problem.

The number 7 is considered to be a lucky number in both Western and Chinese cultures.

The number 7 is a prime number. A happy number is a number that yields a final value of 1, when the individual digits of that number are repeatedly squared and added up. Starting with 7, we have 7^2= 49, and subsequently 4^2 + 9^2 = 16 + 81 = 97. Then, 9^2 + 7^2 = 81 + 49 = 130. Next, 1^2 + 3^2 + 0^2 = 1 + 9 = 10. Finally, 1^2 + 0^2 = 1.

The Millennium Prize Problems consist of seven challenging mathematical problems. The list was compiled by the Clay Mathematics Institute in the year 2000. One of the problems, the Poincaré conjecture, was solved by Russian mathematician Grigori Perelman in 2003, but he declined the USD 1 million prize money.

The remaining six problems are the Birch and Swinnerton-Dyer conjecture, the Hodge conjecture, the Navier-Stokes existence and smoothness problem, the P versus NP problem, the Riemann hypothesis, and the Yang-Mills existence and mass gap problem.

Answer:
**6**

A perfect number can be expressed as the sum of its positive divisors. The first few perfect numbers are 6, 28, 496, and 8128. Observe that 6 = 1 + 2 + 3.

In addition, 6 = 1 + 2 + 3 = 1*2*3, making it the only integer that can be expressed as the sum and the product of 1, 2, and 3.

A perfect number can be expressed as the sum of its positive divisors. The first few perfect numbers are 6, 28, 496, and 8128. Observe that 6 = 1 + 2 + 3.

In addition, 6 = 1 + 2 + 3 = 1*2*3, making it the only integer that can be expressed as the sum and the product of 1, 2, and 3.

Answer:
**10**

The decimal (base 10) numeral system is the most commonly-used numeral system. Other less-commonly used numeral systems include binary (base 2), octal (base 8), and hexadecimal (base 16).

The number 10 can be expressed as the sum of the first four factorials, where 10 = 0! + 1! + 2! + 3! = 1 + 1 + 2 + 6. In addition, the number 10 is also the sum of the first four positive integers, where 10 = 1 + 2 + 3 + 4. The number 10 can also be expressed as the sum of the first three consecutive prime numbers, where 10 = 2 + 3 + 5.

The decimal (base 10) numeral system is the most commonly-used numeral system. Other less-commonly used numeral systems include binary (base 2), octal (base 8), and hexadecimal (base 16).

The number 10 can be expressed as the sum of the first four factorials, where 10 = 0! + 1! + 2! + 3! = 1 + 1 + 2 + 6. In addition, the number 10 is also the sum of the first four positive integers, where 10 = 1 + 2 + 3 + 4. The number 10 can also be expressed as the sum of the first three consecutive prime numbers, where 10 = 2 + 3 + 5.

Answer:
**3**

The third positive integer, 3, is also the first odd prime number. With the exception of 2, all the other prime numbers are odd.

Both pi (3.142...) and Euler's number (2.718...) are irrational. For practical applications, the number 3 is used to provide rough approximations to the problems at hands.

For example, the area, A, of a circle is given by the formula A = pi*r^2, where r is the radius of the circle. Instead of using the exact value of 3.142... to calculate the area, one can use 3 to replace pi, giving an area of 3*2*2 = 12 square units, for a circle with a radius of 2 units. However, do keep in mind that this answer of 12 square units is a very crude approximation, to the actual answer of 12.57 square units.

The third positive integer, 3, is also the first odd prime number. With the exception of 2, all the other prime numbers are odd.

Both pi (3.142...) and Euler's number (2.718...) are irrational. For practical applications, the number 3 is used to provide rough approximations to the problems at hands.

For example, the area, A, of a circle is given by the formula A = pi*r^2, where r is the radius of the circle. Instead of using the exact value of 3.142... to calculate the area, one can use 3 to replace pi, giving an area of 3*2*2 = 12 square units, for a circle with a radius of 2 units. However, do keep in mind that this answer of 12 square units is a very crude approximation, to the actual answer of 12.57 square units.

Answer:
**2**

The first few prime numbers are 2, 3, 5, 7, and 11, making 2 the smallest and the only even prime number.

In mathematics, parity refers to the classification of integers in either one of the two groups of odd or even. All even integers are divisible by 2. For example, 6/2 = 3. In contrast, 7 is an odd integer because the division operation of 7/2 gives a remainder of 1.

In the field of number theory, the irrationality of the square root of 2 can be proven by using a common proof technique called contradiction.

The first few prime numbers are 2, 3, 5, 7, and 11, making 2 the smallest and the only even prime number.

In mathematics, parity refers to the classification of integers in either one of the two groups of odd or even. All even integers are divisible by 2. For example, 6/2 = 3. In contrast, 7 is an odd integer because the division operation of 7/2 gives a remainder of 1.

In the field of number theory, the irrationality of the square root of 2 can be proven by using a common proof technique called contradiction.

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