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Answers
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Q01. Amicable number piars
- Euler discovered 59 amicable pairs.
- Only 8 pairs were found between 1 and 20000 on a Vax computer.
- Where are the other 51 pairs ?
- Reference : p48 of Computer Mathematics by Dr. Shih.
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Study subject
Amicable number piars
Examples
- 1. Prove that 220 and 284 are the first amicable number piars.
- Factors of 220 : 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110.
- Sum of factors : 1+ 2+ 4+ 5+ 10+ 11+ 20+ 22+ 44+ 55+ 110 = 284.
- Factors of 284 : 1, 2, 4, 71, 142.
- Sum of factors : 1+ 2+ 4+ 71+ 142 = 220.
- Hence 220 and 284 are amicable pairs.
- 2. If 1184 and n are an amicable number pairs, find b.
- Factors of 1184 : 1, 2, 4, 8, 16, 32, 37, 74, 148, 296, 592.
- Sum of Factors = 1+ 2+ 4+ 8+ 16+ 32+ 37+ 74+ 148+ 296+ 592 = 1210.
- Hence n = 1210
- Verify :
- Factors of 1210 : 1, 2, 5, 10, 11, 22, 55, 110, 121, 242, 605.
- Sum of factors = 1+ 2+ 5+ 10+ 11+ 22+ 55+ 110+ 121+ 242+ 605 = 1184
- Hence 1184 and 1210 are amicable number pairs.
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Q02. Euler introudced e, i, pi into mathematical field
- e = 1 + 1 + 1/2! + 1/3! + 1/4! + 1/5! + ......
- i = Sqr(-1) and i^2 = -1.
- pi = 3.141592.....
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Q03. Euler formula : e^(i*x) = cos(x) + i*sin(x)
- Series of e^x = 1 + x + x^2/2! + x^3/3! + x^4/4! + ......
- Series of sin(x) = x - x^3/3! + x^5/5! - .....
- Series of cos(x) = 1 - x^2/2! + x^4/4! - .....
- e^(i*x) = 1 + i*x + (i*x)^2/2! + (i*x)^3/3! + (i*x)^4/4! + ....
- Since i^2 = -1, i^3 = -i, i^4 = 1, i^5 = i, .....
- Hence e^(i*x) = (1 - x^2/2! + x^4/4! - ....) + i*(x - x^3/3! + x^5/5! - ....)
- Hence e^(i*x) = cos(x) + i*sin(x).
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Q04. Prove that e^(i*pi) = -1
- e^(i*pi) = cos(pi) + i*sin(pi).
- cos(pi) = -1 and sin(pi) = 0.
- Hence e^(i*pi) = -1.
- This expression includes the most significant symbols -, 1, e, i, pi.
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Q05. i, i^2, i^3, i^4
- i = Sqr(-1)
- i^2 = -1
- i^3 = -i
- i^4 = +1
- i^5 = +i
- i^6 = -1
- i^7 = -i
- i^8 = +1
- Hence i^(4*n) = 1
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Q06. Answer
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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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