Minggu 7Wah Mong WehMathematics DepartmentIPGM KSAH

The Fundamental Theorem of Arithmetic

Teorem Asas Arithmetik(TAR)

The Fundamental Theorem of ArithmeticEvery natural number can be written as the product of primes and (apart from the order in which the primes occur) this factorisation is unique.

Teorem Asas ArithmetikTAR memberitahu kita hubungan antara nombor komposit dengan nombor perdana.Teorem ini menyatakan bahawa: Nombor asli boleh difaktorkan secara unik sebagai hasil darab nombor perdana dalam cara yang unik

The Fundamental Theorem of Arithmetice.g 60 = 2 x 2 x 3 x 5

60 = 2 x 3 x 2 x 5

60 = 3 x 2 x 2 x 5

TAR memberitahu kita bahawa tiada cara untuk memfaktorkan 60 dalam nombor perdana yang lain selain daripada yang ditulis di atas. Pemfaktoran ini adalah unik. Susunan faktor adalah tidak penting

Pokok FaktorCara menulis sebarang nombor komposit dalam bentuk hasil darab nombor perdana- Pokok FaktorCari pemfaktoran perdana bagi (a) 60 (b) 380 ( c) 278 dan tulis jawapan dalam bentuk notasi eksponen

PRIME NUMBERS

A natural number p is called a prime number if and only if p is greater than 1 and the only natural numbers which are factors of p are 1 and p

Nombor PerdanaNombor Perdana adalah integer positif p dimana p>1 jika ia hanya boleh dibahagi oleh nombor positif 1 dan p (diri sendiri)Nombor Perdana adalah nombor yang mempunyai hanya DUA faktor sahaja

PRIME NUMBERS

The prime numbers less than 20 are 2,3,5,7,11,13,17, and 19

TheoremIf a given prime number p is not a factor of n, then no multiple of p is a factor of n

Criterion for finding prime numbersWhether a natural number n is a prime concerns the determination of the largest prime by which we need divide n

Example

Is 251 a prime ?

Determination of PrimesTo determine whether a natural number n is a prime: Divide n by the successive primes 2, 3, 5, p, where p is the largest prime such that p m and m is given by (m-1)2 < n m2If none of these primes is a factor of n, then n is a prime

The sieve of Eratosthenesweek5\The Sieve of Eratosthenes.doc

Factor testsTests to decide whether 2 or 5 is a factor of a natural number n( in decimal notation)

Factor tests If the units digit is 0,2,4,6, or 8 , then 2 is a factor of n.

If the units digit is 0 or 5, then 5 is a factor of n

Factor testsFactor tests can be proved by means of congruence.

Conjectures about primesIs there a formula which will yield successive primes ?

Conjectures about primes is a prime for n=1 to n =40

is a prime for n=1 to n=79

Conjectures about primesTwin primes are two primes whose different is 2. e.g. 11 and 13 Is there an infinite number of primes?

Thank You

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