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# Perfect and Not-So-Perfect Numbers Quiz

### Here is a quiz on perfect numbers and related interesting groups of numbers.

A multiple-choice quiz by looney_tunes. Estimated time: 6 mins.

Author
Time
6 mins
Type
Multiple Choice
Quiz #
277,744
Updated
Dec 03 21
# Qns
10
Difficulty
Tough
Avg Score
6 / 10
Plays
746
Question 1 of 10
1. A PERFECT NUMBER is one which is the sum of its proper positive factors (the integers smaller than itself which divide evenly into the number). The smallest perfect number is 6; you can show that it is perfect because its factors are 1, 2, 3 and 6. That means its proper factors are 1, 2 and 3. The sum of its proper factors is 1+2+3=6. What is the next smallest perfect number? Hint

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Question 2 of 10
2. Another way of defining a perfect number is to say that the sum of ALL its factors is equal to twice the original number (shown as 2n). An ALMOST PERFECT NUMBER (also called a slightly defective number) has a sum of all its factors equal to 2n-1 (one less than twice the original number). Which of the following numbers is almost perfect? Hint

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Question 3 of 10
3. A QUASIPERFECT NUMBER is one for which the sum of all its factors is equal to 2n+1 (one more than the original number). Have any quasiperfect numbers been found?

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Question 4 of 10
4. A MULTIPERFECT NUMBER (also called a pluperfect number) is one for which the sum of all its positive factors is equal to a multiple of the number itself. 'Normal' perfect numbers have a sum of 2n, and are referred to as being 2-perfect. What is the smallest 3-perfect number, for which the sum of its factors is 3 times the number itself? (Hint: 3-perfect is a clue) Hint

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Question 5 of 10
5. Next let's consider HYPERPERFECT NUMBERS. For a k-hyperperfect number (n), there is an integer k that means the sum of the factors of n is equal to [n(k+1)-1]/k + 1. For k = 2, which of the following is a 2-hyperperfect number? (Hint: Put 2 into the formula where k is written; test each number by putting it where n is and calculating both the resulting value and the sum of the factors of n.) Hint

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Question 6 of 10
6. Is a hyperperfect number also a perfect number? (Hint: Try seeing if a few different values of k result in an expression that can be simplified to 2n.) Hint

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Question 7 of 10
7. A UNITARY PERFECT NUMBER is the sum of its unitary factors. To demonstrate what this means, consider any number, a, for which b is a proper factor (a factor smaller than a). This means that a/b is an integer. If that integer and b have no common factor greater than 1, b is said to be a unitary factor of a.

Consider 12, which has factors (among others) of 4 and 6. Since 12/4 = 3, and there is no common factor of 3 and 4 greater than 1, 4 is a unitary factor of 12. Since 12/6 = 2, and 2 and 6 do have a common factor greater than 1 (2 is the common factor), 6 is NOT a unitary factor of 12.

Which of the following is a unitary perfect number?
Hint

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Question 8 of 10
8. A SEMIPERFECT NUMBER is the sum of some or all of its proper factors (factors smaller than the number itself). 12 is semiperfect because you can take some of its proper factors and add them to get 12: 2+4+6=12. Which of the following is NOT semiperfect? Hint

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Question 9 of 10
9. An ABUNDANT NUMBER (one for which the sum of its proper factors is greater than itself) which is not also semiperfect (the sum of some or all proper factors equals the number itself) is called a WEIRD NUMBER. What is the smallest weird number? (Hint: start with the smallest, and check the given numbers until you find one which is both abundant and semiperfect.) Hint

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Question 10 of 10
10. 6, the smallest perfect number, is also a PRACTICAL NUMBER. This means that all the integers less than it can be represented as the sum of distinct factors of the number. (Distinct means that each factor can only be used once in the sum.)
1 = 1; 2 = 2; 3 = 3 (or 2+1); 4 = 3+1; 5=3+2

Which of these is also a practical number?
Hint

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Quiz Answer Key and Fun Facts
1. A PERFECT NUMBER is one which is the sum of its proper positive factors (the integers smaller than itself which divide evenly into the number). The smallest perfect number is 6; you can show that it is perfect because its factors are 1, 2, 3 and 6. That means its proper factors are 1, 2 and 3. The sum of its proper factors is 1+2+3=6. What is the next smallest perfect number?

A list of perfect numbers can be found at the On-Line Encyclopedia of Integer Sequences (OEIS). Perfect numbers are OEIS sequence A000396.

If the sum of the proper factors is less than the original number, that number is said to be deficient. If the sum is greater than the original number, the number is said to be abundant. 8 is deficient (1+2+4=7), while 12 and 24 are abundant (1+2+3+4+6=16 which is greater than 12; 1+2+3+4+6+8+12=36, which is greater than 24).
2. Another way of defining a perfect number is to say that the sum of ALL its factors is equal to twice the original number (shown as 2n). An ALMOST PERFECT NUMBER (also called a slightly defective number) has a sum of all its factors equal to 2n-1 (one less than twice the original number). Which of the following numbers is almost perfect?

Answer: all of them

1: The only factor is 1, which is equal to 2*1-1.
2: The factors are 1 and 2; their sum is 3, which is equal to 2*2-1.
4: The factors are 1, 2 and 4; their sum is 7, which is equal to 2*4-1.

The only known odd almost perfect number is 1. All known almost perfect numbers can be written in the form 2^n.
3. A QUASIPERFECT NUMBER is one for which the sum of all its factors is equal to 2n+1 (one more than the original number). Have any quasiperfect numbers been found?

While none of these hypothetical numbers has yet been found, it has been shown that, if one exists, it must be an odd square number with at least 39 digits, and must have at least 7 distinct prime factors. They're still looking.
4. A MULTIPERFECT NUMBER (also called a pluperfect number) is one for which the sum of all its positive factors is equal to a multiple of the number itself. 'Normal' perfect numbers have a sum of 2n, and are referred to as being 2-perfect. What is the smallest 3-perfect number, for which the sum of its factors is 3 times the number itself? (Hint: 3-perfect is a clue)

Factors of 96 are 1,2,3,4,6,8,12,16,24,32,48,96. Their sum is 252, which is not a multiple of 96.

Factors of 120 are 1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120. The sum of these numbers is 360, which is 120*3.

A list of multiperfect numbers can be found at OEIS sequence A007539. 30240 is the smallest 4-perfect number. The smallest 5-perfect number is 14182439040.
5. Next let's consider HYPERPERFECT NUMBERS. For a k-hyperperfect number (n), there is an integer k that means the sum of the factors of n is equal to [n(k+1)-1]/k + 1. For k = 2, which of the following is a 2-hyperperfect number? (Hint: Put 2 into the formula where k is written; test each number by putting it where n is and calculating both the resulting value and the sum of the factors of n.)

21 is 2-hyperperfect, as the sum of its factors is 1+3+7+21=32 and [21(2+1)-1]/2 + 1 = 32.

As 24 and 28 are both even numbers, multiplying by 3 then subtracting 1 will give an odd number, so division by 2 cannot give an integer.

For 27, 1+3+9+27=40 and [27(2+1)-1]/2+1 = 41. Since these numbers are different, 27 is not 2-hyperperfect.

The sequence of hyperperfect numbers can be found at OEIS sequence A034897.
6. Is a hyperperfect number also a perfect number? (Hint: Try seeing if a few different values of k result in an expression that can be simplified to 2n.)

Answer: Rarely, but it is possible under some conditions

For a k-hyperperfect number (n), there is an integer k that means the sum of the factors of n is equal to [n(k+1)-1]/k + 1.

For a perfect number, the sum of its factors is 2n.

For a hyperperfect number with k=1, the expression [n(k+1)-1]/k + 1 simplifies to 2n, which is a perfect number. For most values of k, however, the k-hyperperfect number is not perfect.
7. A UNITARY PERFECT NUMBER is the sum of its unitary factors. To demonstrate what this means, consider any number, a, for which b is a proper factor (a factor smaller than a). This means that a/b is an integer. If that integer and b have no common factor greater than 1, b is said to be a unitary factor of a. Consider 12, which has factors (among others) of 4 and 6. Since 12/4 = 3, and there is no common factor of 3 and 4 greater than 1, 4 is a unitary factor of 12. Since 12/6 = 2, and 2 and 6 do have a common factor greater than 1 (2 is the common factor), 6 is NOT a unitary factor of 12. Which of the following is a unitary perfect number?

Proper factors of 6 are 1, 2 and 3.
6/1 = 6, and 1 & 6 have no common factor greater than 1, so 1 is a unitary factor.
6/2 = 3, and 2 & 3 have no common factor greater than 1, so 2 is a unitary factor of 6.
6/3 = 2. and 3 & 2 have no common factor greater than 1, so 3 is a unitary factor of 6.
The sum of the unitary factors is 1+2+3=6, so 6 is a unitary perfect number.

Similar calculations for the other numbers show that they are NOT unitary perfect numbers.

The first few unitary perfect numbers are 6, 60, 90, 87360, 14636194618645856256000. Then they start getting big. (OEIS sequence A002827)
8. A SEMIPERFECT NUMBER is the sum of some or all of its proper factors (factors smaller than the number itself). 12 is semiperfect because you can take some of its proper factors and add them to get 12: 2+4+6=12. Which of the following is NOT semiperfect?

For 6, proper factors are 1,2,3. 1+2+3=6, so 6 is semiperfect.
For 15, proper factors are 1,3,5. 1+3+5=9, so 15 is NOT semiperfect.
For 18, proper factors are 1,2,3,6,9. 9+3+6=18, so 18 is semiperfect.
For 20, proper factors are 1,2,4,5,10. 10+5+4+1=20, so 20 is semiperfect.

The smallest odd semiperfect number is 945. All multiples of a semiperfect number are also semiperfect. (OEIS sequence A005835)
9. An ABUNDANT NUMBER (one for which the sum of its proper factors is greater than itself) which is not also semiperfect (the sum of some or all proper factors equals the number itself) is called a WEIRD NUMBER. What is the smallest weird number? (Hint: start with the smallest, and check the given numbers until you find one which is both abundant and semiperfect.)

12 is abundant, since 1+2+3+4+6=16, but it is also semiperfect, as 6+4+2=12.

70 has proper factors of 1,2,5,7,10,14,35; their sum is 74, so 70 is abundant.
There is no combination of these that can be used to get a sum of 70, so 70 is not semiperfect. By definition, 70 is weird.

836 is the second-smallest weird number. 120 is abundant 1+2+3+4+5+6+8+10+12+15+20+24+30+40+60=240), but it is also semiperfect (60+40+20=120).

(OEIS sequence A006037)
10. 6, the smallest perfect number, is also a PRACTICAL NUMBER. This means that all the integers less than it can be represented as the sum of distinct factors of the number. (Distinct means that each factor can only be used once in the sum.) 1 = 1; 2 = 2; 3 = 3 (or 2+1); 4 = 3+1; 5=3+2 Which of these is also a practical number?

Answer: all of these

This takes a lot of time to show, but is not difficult. Here is the work for 12:
1 = 1; 2 = 2; 3 = 3; 4 = 4; 5 = 4+1; 6 = 4+2; 7 = 4+3; 8 = 4+3+1; 9 = 4+3+2; 10 = 6+4; 11 = 6+4+1.

Practical numbers are actually MUCH more common than the other groups of numbers at which we have been looking. All even perfect numbers are practical, as are all powers of 2. (OEIS sequence A005153)

If you have found these interesting, you might like to explore amicable numbers, friendly numbers, sociable numbers, solitary numbers and sublime numbers - who knew numbers had so much personality!
Source: Author looney_tunes

This quiz was reviewed by FunTrivia editor crisw before going online.
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