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

Quiz Answer Key and Fun Facts

Answer:
**28**

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).

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).

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.

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.

Answer:
**No **

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.

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.

Answer:
**120**

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.

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.

Answer:
**21**

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.

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.

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.

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.

Answer:
**6**

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)

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)

Answer:
**15**

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)

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)

Answer:
**70**

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)

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)

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!

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!

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

Any errors found in FunTrivia content are routinely corrected through our feedback system.

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