# Chapter 7: Factorization and fundamental theorem of arithmetic (1)

Time：2020-12-2

Prime numbers are numbers like thisp\geq2Its (positive) factor is only1Andp。 Integers that are not prime numbersm\geq2It’s called summation.

Prime numbers are characterized by such divisibility of numbers that they can only be1And they themselves divide the definition of this property.

Theorem: letpIt’s a prime. SupposepDivisible productab, thenpto be divisible byaOr divideb(orpBoth divisionaAlso divideb)

Proof: knownpDivisible productab, ifpto be divisible bya, then it is proved that it has been completed, so it can be assumed thatpDo not dividea。 Now think about itgcd(p,a)What is it. Since it is divisiblepIt is1orpIt’s also divisiblea, assumingpDo not dividea, sogcd(p,a)nop, sogcd(p,a)Must be equal to1

The following applicationpAndaThe linear equation theorem points out that the equation can be solvedpx+ay=1Integer solution ofxAndy。 (pay attention to the use of factsgcd(p,a)=1）Now, multiply both sides of the equationbhave topxb+aby=bObviously,pto be divisible bypbxDue topto be divisible byab,pto be divisible byaby。 soPDivisional sumpbx+abySo thatpto be divisible byb