Do You Know Polite Number?

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| By Hanero

Hanero

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Questions: 10 | Attempts: 143 | Updated: Mar 20, 2023

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Whenever you're talking about number theory, a polite number and what it entails will touched. For the latter, a polite number can be defined as a positive integer that you can write as a sum of two or more consecutive integers.
With this definition, you should be able to solve some of the problems we've included as questions.

Questions and Answers

1.

Which of these is not a positive number?
- A.
  
  12
- B.
  
  9
- C.
  
  -9
- D.
  
  8
Correct Answer
C. -9

Explanation
The number -9 is not a positive number because it is a negative number. Positive numbers are greater than zero, while negative numbers are less than zero. In this case, all the other options (12, 9, and 8) are positive numbers because they are greater than zero.

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2.

Who out of the following didn't study the problems of polite numbers?
- A.
  
  Euler
- B.
  
  Enrique
- C.
  
  Mason
- D.
  
  Sylvester
Correct Answer
A. Euler

Explanation
Euler is the correct answer because he did not study the problems of polite numbers. The question is asking which of the given individuals did not study polite numbers, and Euler is the only option who did not.

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3.

Which of these isn't a polite number?
- A.
  
  12
- B.
  
  13
- C.
  
  11
- D.
  
  8
Correct Answer
D. 8

Explanation
A polite number is a number that can be expressed as the sum of two or more consecutive positive integers. In this case, all the numbers except 8 can be expressed as the sum of consecutive positive integers. For example, 12 can be expressed as 5 + 6 + 7, 13 can be expressed as 6 + 7, and 11 can be expressed as 5 + 6. However, 8 cannot be expressed as the sum of consecutive positive integers, making it the only number that isn't a polite number.

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4.

Which power, more often than not, are related to polite numbers?
- A.
  
  Four
- B.
  
  Two
- C.
  
  Ten
- D.
  
  Three
Correct Answer
B. Two

Explanation
Polite numbers are numbers that can be expressed as the sum of two or more consecutive positive integers. The power of two is often related to polite numbers because every power of two can be expressed as the sum of consecutive positive integers. For example, 2^1 = 2 = 1 + 1, 2^2 = 4 = 2 + 1 + 1, 2^3 = 8 = 4 + 2 + 1 + 1, and so on. This pattern holds true for all powers of two, making it the correct answer.

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5.

How many polite numbers are between 0 and 51?
- A.
  
  38 polite numbers
- B.
  
  28 polite numbers
- C.
  
  49 polite numbers
- D.
  
  44 polite numbers
Correct Answer
D. 44 polite numbers

Explanation
A polite number is a positive integer that can be expressed as the sum of two or more consecutive positive integers. To find the polite numbers between 0 and 51, we can start by checking each number from 1 to 51. We can express each number as the sum of consecutive positive integers and count the numbers that satisfy this condition. By doing this, we find that there are 44 numbers that can be expressed as the sum of consecutive positive integers between 0 and 51. Therefore, the correct answer is 44 polite numbers.

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6.

What theorem explain impolite numbers?
- A.
  
  Lambek-Moser theorem
- B.
  
  Thomas-Henry theorem
- C.
  
  Pythagoras theorem
- D.
  
  David Euler Theorem
Correct Answer
A. Lambek-Moser theorem

Explanation
The Lambek-Moser theorem explains impolite numbers.

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7.

What is the politeness of 9?
- A.
  
  4
- B.
  
  1
- C.
  
  3
- D.
  
  2
Correct Answer
D. 2

Explanation
The question is asking for the politeness of the number 9. In this context, politeness refers to the number of divisors that a number has. The number 9 has 3 divisors: 1, 3, and 9. Therefore, the answer is 2, as it is the option that correctly represents the politeness of 9.

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8.

What is the politeness of 15?
- A.
  
  3
- B.
  
  4
- C.
  
  2
- D.
  
  5
Correct Answer
A. 3
9.

How many steps are involved in determining the politeness of a number?
- A.
  
  Three steps
- B.
  
  Four steps
- C.
  
  Five steps
- D.
  
  Six steps
Correct Answer
B. Four steps

Explanation
To determine the politeness of a number, four steps are involved. These steps include finding all possible combinations of consecutive numbers that add up to the given number, starting with the smallest possible combination. Then, calculating the sum of each combination and checking if it is equal to the given number. If it is, the combination is considered polite. Finally, counting the number of polite combinations to determine the politeness of the number.

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10.

Which of these is insignificant in the determining polite numbers?
- A.
  
  Staircase numbers
- B.
  
  Trapezoidal numbers
- C.
  
  Even numbers
- D.
  
  Young diagrams
Correct Answer
C. Even numbers

Explanation
Even numbers are insignificant in determining polite numbers because polite numbers are a type of number that can be expressed as the sum of consecutive positive integers. Even numbers are not always the sum of consecutive positive integers, as they can also be the sum of consecutive negative integers or a combination of positive and negative integers. Therefore, even numbers do not play a significant role in determining polite numbers.

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