# MATH 1 - Sections 16 & 17 Handout for Chapter 5

# MATH 1 - Sections 16 & 17 Handout for Chapter 5

Chapter 5 deals with Number Theory . Read page 221. We are
mainly

concerned with in this chapter. You should become conversant with

• b | a means b devides a,

• means b does not divide a

• factor, divisor, multiple , factorization (p.222), prime number , compos-

ite number, Sieve of Eratothenes (p.223-224)

• Divisibility tests (Table 2 on p.225), Definition of Theorem(p.225)

and the Fundamental Theorem of Arithmetic

• Obtaining the unique factorization by direct computation

• The theorem on the infinitude of primes

• The search for large primes, Mersenne numbers, GIMPS, Fermat Num-

bers, Euler formula, Escott formula

• **HW 5.1:** 1−24, Odd numbered exercises from 27 to 79, 80 and 81

• In section 5.2 we study about Perfect numbers , Deficient and Abundant

numbers, Amicable numbers, Weird and Dull numbers(?!), The man

who knew infinity, Definition of Conjecture(p.2, p.6), Goldbach's

conjecture, the Twin prime conjecture, Fermat's Last Theorem and

Andrew Wiles

• **HW 5.2:** 1−10, Odd numbered exercises from 11−43, 54, 56, 58-60.

• 5.3 treats the Greatest Common Factor (GCF), the Least CommonMul-

tiple( LCM ), Relatively prime numbers, Three methods for finding the

GCF and Three methods for finding the LCM

• **HW 5.3:** 1−10, 13, 15, 17, 21, 23, 25, 27, 33, 35, 37, 41, 43, 45, 47,

49, 50, 51, 55, 65, 67, 69.

• In 5.1 we discussed the formula for the number of all
divisors. If n =

then N has (a+1)(b+1)(c+1) divisors. For example, in #59

(p.232), 48 = 2^{4}.3 and so 48 has (4 + 1)(1 + 1) = (5) (2) = 10 divisors.

• #72, #73 (p.232): For n = 42 formula gives
(42)^{2} − (42) +

41 = 1763 = 43.41 which is composite. For n = 43 the formula gives

(43)^{2}−(43)+41 = 1847 which can be verified to be a prime by checking

for prime factors below 43 since 43 is close to
).

• You have to study 5.2 carefully to understand proper divisors and find-

ing the sum of proper divisors. That can be used to classify every

number as perfect, abundant or deficient. Why are all prime numbers

deficient?

• 5.4 treats the well known Fibonacci Sequence
and the

Golden Ratio
. HW for 5.4: 1-6, 15, 17, 20, 31.

• We will briefly discuss Modern Cryptography (p.249 − 256) and Magic

Squares (p.265 − 270).

• As a review of your mastery of the concepts we discussed solve from

Chapter 5 test: 1-16, 18, 19.

• Using a program called Mathematica (created by Stephen Wolfram ) I

could establish that **5.7.11.13.17.19.23.29.31 = 33,426,748,355** is

an abundant number since the sum of its proper divisors is calculated

to be **33,459,293,245**.

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