Exercise · ZW-GREATEST-COMMON-DIVISOR-006 · ★★☆ Intermediate
Compute \(\gcd(1023,63)\), that is, \(\gcd(2^{10}-1,\,2^{6}-1)\).
Choose one option.
Show answer
- D. 3
Explanation
With \(1023=16\cdot63+15\), \(\gcd(1023,63)=\gcd(63,15)=\gcd(15,3)=3\). This matches \(2^{\gcd(10,6)}-1=2^2-1=3\).
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-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)?
- ZW-GREATEST-COMMON-DIVISOR-009 How many integers n with 1≤ n≤100 satisfy gcd(n,100)=1?
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