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Mathematics Dictionary
Dr. K. G. Shih
Properties of Factors of Numbers
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Q01. Factors of numbers

  • Definition
    • A number m is divisible by number n, then number n is a factor of number m.
    • Example : 6 is divisible by 3 and then 3 is a factor of 6.
  • Notes
    • Number 1 is a factor of all real numbers.
    • Number itself is also a factor of the number.
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Q02. Properties of numbers
Defintion : Sum of factors of a number without number itself has following properties
  • Abundunt number.
    • Defintion : Sum of factors of a number is greater than number itself.
    • Example : factors of 30 are 1, 2, 3, 5, 6, 10, 15.
    • Sum of factors = 1 + 2 + 3 + 5 + 6 + 10 + 15 = 41 which is greater than 30.
    • Hence 30 is an abundunt number.
  • Deficient number.
    • Defintion : Sum of factors of a number is less than number itself.
    • Example : factors of 27 are 1, 3, 9.
    • Sum of factors = 1 + 3 + 9 = 13 which is less than 27.
    • Hence 27 is a deficient number.
  • Perfect number.
    • Defintion : Sum of factors of a number is equal to number itself.
    • Example : factors of 6 are 1, 2, 3.
    • Sum of factors = 1 + 2 + 3 = 6 which is equal to 6.
    • Hence 6 is the first perfect number.
  • Prime number.
    • Defintion : The number can only divided by 1 or itself.
    • A prime number can only have two factors. e.g. factors of 11 are 1 and 11.
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Q03. Amicable numbers
  • Study | Amicable number pairs
    • Example 1. What are amicable number pairs ?
    • Example 2. Prove that 220 and 284 are amicable pairs.
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Q04. Perfect numbers
  • Study | Perfect numbers.
    • Example 1. What is perfect number ?
    • Example 2. Prove that 28 is a perfect number.
    • Example 3. Find the 3rd perfect number by using properties of perfect number.
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Q05. Prime numbers
Twin prime numbers
  • Defintion : Two prime numbers only separated one number.
  • Example : 11 and 13 are twin prime numbers.
  • Example : 3, 5, 7 are three prime numbers. Between prime number only one number which are not prime. This is the only case.
Special properties
  • Sum of two prime numbers is even except number 2.
  • There is no proof.
  • Only one prime number is even. It is number 2.
  • Reference : See Goldbach's projective in MD2002 01 21.
Special positions in patterns : Prime numbers in 1st column or 5th column except 2 and 3 as shown in the following pattern.
  • 01 02 03 04 05 06
  • 07 08 09 10 11 12
  • 13 14 15 16 17 18
  • 19 20 21 22 23 24
  • 25 26 27 28 29 30
  • 31 32 33 34 35 36
  • 37 38 39 40 41 42
  • 43 44 45 46 47 48
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Q06. Prime factors of numbers
  • Definition : The factors of numbers can be express by prime numbers.
  • Example 1 : express 15 as prime factor.
    • 15 = 3*5. Numbers 3 and 5 are prime numbers. Hence it is prime factor.
  • Example 2 : express 8 as prime factor.
    • 8 = 2*4. Since 4 is not prime number. Hence this is not a prime factor.
    • Can we express 8 as prime factor ?
    • Yes. 8 = 2*2*2 is prime factors.
  • Example 3 : express 165 as prime factors
    • 165 = 3*5*11 is prime factor form.
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Q07. Answer
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Q08. Answer
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Q09. Answer
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Q10. Answer
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