{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# gcdproofkey - McCombs Math 81 Working with gcd(a,b...

This preview shows pages 1–2. Sign up to view the full content.

McCombs Math 81 Working with gcd( a , b ) 1 Important Theorems: Theorem: Given , a b ! Z , with a and b not both 0. If ( ) gcd , d a b = , then there exist , x y ! Z such that d ax by = + . Theorem: Given , a b ! Z , with a and b not both 0. If ( ) gcd , d a b = , then d is the smallest positive integer that can be expressed as the linear combination d ax by = + , where , x y ! Z . Theorem: Integers a and b are relatively prime if and only if there exist , x y ! Z such that 1 ax by + = . Proof Examples: 1. Given , , a b w ! Z . The equation ax by w + = has integer-valued solutions , x y if and only if ( ) gcd , a b w . Part (i): Assume the equation ax by w + = has integer-valued solutions , x y . Let ( ) gcd , d a b = . We know that a kd = and b md = for some integers m and n . ( ) ( ) ax by w kd x md y w + = ! + = So ( ) w d kx my = + . Since k , x , m , and y are all integers, d w . Thus, ( ) gcd , a b w . Part (ii): Assume ( ) gcd , a b w . Let ( ) gcd , d a b = . We know that a kd = and b md = for some integers m and n . We also know that there exist integers m and n such that ma nb d + = . ( ) gcd , a b w w dk ! = for some integer k . So we have ( ) w dk ma nb k kma knb = = + = + . Thus, x mk = and y nk = are integer-valued solutions for the equation ax by w + = . Therefore, the equation ax by w + = has integer-valued solutions , x y if and only if ( ) gcd , a b w .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
McCombs Math 81 Working with gcd(
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 2

gcdproofkey - McCombs Math 81 Working with gcd(a,b...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online