Exercise · ZW-GREATEST-COMMON-DIVISOR-010 · ★★★ Advanced
If \(\gcd(a,b)=1\), what values can \(\gcd(a+b,\,a-b)\) take for \(a\ne b\)?
Choose one option.
Show answer
- D. Only 1 or 2
Explanation
A common divisor \(d\) divides the sum \(2a\) and the difference \(2b\), so \(d\mid 2\gcd(a,b)=2\). It is 2 when \(a,b\) are both odd (3 and 1: \(\gcd(4,2)=2\)) and 1 otherwise (2 and 1: \(\gcd(3,1)=1\)).
Review the concept: Greatest common divisor
The greatest common divisor \(\gcd(a,b)\) of two integers \(a\) and \(b\), not both zero, is the largest integer that divides both.
\[\gcd(a,b)=\max\{d\in\mathbb{Z}:\ d\mid a,\ d\mid b\}\]
More practice problems: greatest common divisor
- ZW-GREATEST-COMMON-DIVISOR-001 Compute gcd(12,18).
- ZW-GREATEST-COMMON-DIVISOR-002 Compute gcd(35,64).
- ZW-GREATEST-COMMON-DIVISOR-003 Compute gcd(0,9).
- ZW-GREATEST-COMMON-DIVISOR-004 Compute gcd(48,180).
- ZW-GREATEST-COMMON-DIVISOR-005 Compute gcd(84,126).
- ZW-GREATEST-COMMON-DIVISOR-006 Compute gcd(1023,63), that is, gcd(2^10-1, 2^6-1).
- ZW-GREATEST-COMMON-DIVISOR-007 For which value of b does gcd(20,b)=4 hold?
- ZW-GREATEST-COMMON-DIVISOR-008 If d=gcd(a,b) with a,b nonzero, what is gcd(a/d, b/d)?