Read Symbol defintion
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Q01 |
- What is perfect number ?
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Q02 |
- Can we have more perfect numbers ?
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Q03 |
- Properties of perfect number
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Q04 |
- Prove 496 is perfect number
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Q05 |
- Prove 8128 is a perfect number
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Q06 |
- Find 8th perfect number
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Q07 |
- Related to properties of number's factors
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Q08 |
- More sample on internet
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Q09 |
- Factors of 6th perfect number
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Q10 |
- Factors of 7th perfect number
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Answers
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Q01. What is perfect number ?
Definition and examples
- Definition : The sum of factors of a number = number itself
- Examples
- [Example] 1, 2 ,3 are factor of 6 and 1 + 2 + 3 = 6
- [Example] 6 is also a factor of 6 but not used in sum
- [Example] Prove that 28 is a perfect number
- Perfecr numbers
- 1st perfect number is 6
- 2nd perfect number is 28
- 3rd peferct number is 496
- 4th perfect number is 8128
- 5th perfect number is 33550336
- 6th perfect number is 8589869056
- 7th perfect number is 137438691328
- 8th perfect number is 2305843008139952128
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Q02. Can we have more perfect numbers ?
First eight can be abtained using programs in chapter 1.
- 4th perfect number is 8128
- 5th perfect number is 33550336
- 6th perfect number is 8589869056
- 7th perfect number : see program 01 13
- 8th perfect number : see program 01 16
Notes
- Note : program 01 13 will give the factors of perfect number
- Note : program 01 16 show how to compute 8th perfect number
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Q03. Properties of perfect number
Answer : Properties are
- Rule 1 : perfect number = 1^3 + 3^3 + 5^3 + .... + (2*n-1)^3
- Rule 2 : perfect number = (2^(n-1))*(2^n-1)
Note and example
- Note : first property is not applied to 6
- [Example] Prove that 28 is perfect number
- Rule 1 : 1^3 + 3^3 = 1 + 27 = 28
- Rule 2 : (2^(3-1))*(2^3-1) = (2^2)*(8-1) = 4*7 = 28
- Hence 28 is a perfect number
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Q04. Porve 496 is perfect number
Use rule 1 and rule 2 : by trial we get
- Rule 1 : 1^3 + 3^3 + 5^3 + 7^3 = 1 + 27 + 125 + 343 = 496
- Rule 2 : n=5 and (2^(5-1))*(2^5-1) = (2^4)*(32-1) = 16*31 = 496
- Hence 496 is the 3rd fector
Proof by defintion
- Factors of 496 : 1, 2, 4, 8, 16, 31, 62, 124, 248
- Sum of factors = 1+2+4+8+16+31+62+124+248 = 496
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Q05. Prove 8128 is perfect number
Using the properties find the 4th perfect number 8128
- Hint : the 4th number is between 8000 and 9000.
- Rule 1
- N = 496 + 9^3 + 11^3 + 13^3 + 15^3 = 496 + 729 + 1331 + 2197 + 3375 = 8128
- Rule 2
- n = 7 : (2^(7-1))*(2^7 -1) = 64*127 = 8128
Find factors
- Factors : 1, 2, 4, 8, 16, 32, 64, 127, 508, 254, 1016, 2032, 4064
- Sum = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064
- Sum = 8128
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Q06. Find 8th perfect number using QB
Answer : From MD2002 01 16
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Q07. Related to properties of number's factors
Properties
- Deficient number .... if sum of factors is less than number
- Perfect number ...... if sum of factors is equal to number itself
- Abundunt number ..... if sum of factors is greater than number
Example
- Factors of 28 : 1, 2, 4, 7, 14, 28
- 1 + 2 + 4 + 7 + 14 = 28
- 28 is perfect number
- Factors of 29 : 1, 29
- Sum = 1
- 29 is deficient number (a prime number)
- Factors of 30 : 1, 2, 3, 5, 10, 15
- Sum = 1 + 2 + 3 + 5 + 10 + 15 = 36
- 36 is abundunt number
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Q08. More samples on internet
Sample to find factors of perfect number
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Q09. Find factors of 6th perfect number
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Topic Factors of 7th perfect number
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Q10. Factors of 7th perfect number
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