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Mathematics Dictionary
Dr. K. G. Shih
Amicable Numbers


  • Q01 | - Amicable numbers
  • Q02 | - prove that 220 and 284 are amicable numbers
  • Q03 | - prove that 1184 and 1210 are amicable numbers
  • Q04 | - Prove that 2620 and 2924 are amicable pairs
  • Q05 | - References

    • Q01. Amcable numbers

    Definition
    • The sum of factors of 1st number = 2nd number
    • The sum of factors of 2nd number = 1st number
    • Then these two numbers are called amicable pairs
    Example : Prove that 220 and 284 are amicable pairs
    • Sum of factors 284
      • Sum = 1 + 2 + 4 + 71 + 142 = 220
    • Sume of factors 220
      • Sum = 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284
    More amicable number pairs between 1000 and 10000 are
    • 1184 and 1210
    • 2620 and 2924
    • 5052 and 5564
    • 6232 and 6368
    History of amicable numbers
    • In 1636, Fermat found pairs of 17296 and 18416
    • In 1638, Descartes found pairs of 9363584 and 9437056
    • In 1747, there were 30 pairs
    • In 1866, there were 60 pairs
    • In 1946, there were 390 pairs
    • in 1993, there were 3340 pairs (number less than 10^11)

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    Q02. Prove that 220 and 284 are amicable pairs

    Proof :
    • Factors of 220 :
      • 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, 220
      • Sum the factors without 220
      • Sum = 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284
    • Factors of 284
      • 1, 2, 4, 71, 142, 284
      • Sum the factors without 284
      • Sum = 1 + 2 + 4 + 71 + 142 = 220
    • Hence 220 and 284 are amicable pairs

    Go to Begin

    Q03. Prove that 1184 and 1210 are amicable pairs

    Proof
    • Factors of 1184
      • Factors : 1, 2, 4, 8, 16, 32, 37, 74, 148, 296, 592, 1184
      • Sum factors without 1184
      • Sum = 1 + 2 + 4 + 8 + 16 + 32 + 37 + 74 + 148 + 296 + 592 = 1210
    • Factors of 1210
      • Factors : 1, 2, 5, 10, 11, 110, 121, 242, 605, 1210
      • Sum factors without 1210
      • Sum = 1 + 2 + 5 + 10 + 11 + 110 + 121 + 242 + 605 = 1184
    • Hence 1184 and 1210 are amicable pairs
    • This pair was found by a 16 year old boy in 1866

    Go to Begin

    Q4. Prove that 2620 and 2924 are amicable pairs

    Proof
    • Factors of 2620
      • 1, 2, 4, 5, 10, 20, 131, 262, 524, 655, 1310
      • Sum of factors without 2620
      • Sum = 1 + 2 + 4 + 5 + 10 + 20 + 131 + 262 + 524 + 655 + 1310
      • Sum = 2924
    • Factors of 2924
      • Factors : 1, 2, 4, 17, 34, 43, 68, 86, 172, 731, 1462
      • Sum of factors without 2924
      • Sum = 1 + 2 + 4 + 17 + 34 + 43 + 68 + 86 + 172 + 731 + 1462
      • Sum = 2620
    • Hence 2620 and 2924 are amicable pairs

    Go to Begin

    Q5. References

    References :
    • p234-p235 Computer Mathematics by Dr shih
    • Mathematics Dictionary Chapter 1 program 9 and 24
    • Keyword : Euler
    • Keyword : Amicable
    • Subject : Algebra section 3
    • DrMath : Search amicable

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