# sol2 - Math 465 Solution Set 2 1 If(a c = 1 and(b c = 1...

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

Math 465 Solution Set 2 Ae Ja Yee 1. If ( a, c ) = 1 and ( b, c ) = 1, then ( ab, c ) = 1. Solution. Method 1: Suppose ( ab, c ) > 1, namely x | ab and x | c for some x > 1. Since ( a, c ) = 1, x cannot divide a , that is ( x, a ) = 1. However, x | ab , so x | b , which is a contradiction. Method 2: We can write 1 = ax 1 + cy 1 = bx 2 + cx 2 . Thus ax 1 bx 2 = (1 - cy 1 )(1 - cy 2 ) = 1 - c ( y 1 + y 2 - cy 1 y 2 ) , which is equivalent to 1 = abx + cy, where x = x 1 x 2 and y = y 1 + y 2 - cy 1 y 2 . Therefore, ( ab, c ) = 1. 2 2. Show that if ( a, b ) = 1, then ( a - b, a + b ) = 1 or 2. When is the value 2? Solution. Method 1: Let n = a - b , m = a + b , and ( n, m ) = d . Then 2 a = m + n, 2 b = m - n Since ( n, m ) = d , d | 2 a and d | 2 b , that is d is a common divisor of 2 a and 2 b . We now compute (2 a, 2 b ): (2 a, 2 b ) = 2( a, b ) = 2 since ( a, b ) = 1. Therefore, d | 2, that is d = 1 or 2. Method 2: Since gcd is the least positive linear combination, (

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 2

sol2 - Math 465 Solution Set 2 1 If(a c = 1 and(b c = 1...

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

View Full Document
Ask a homework question - tutors are online