GCD from Prime Decomposition/Examples/64 and 81
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Example of Use of GCD from Prime Decomposition
The greatest common divisor of $64$ and $81$ is:
- $\gcd \set {64, 81} = 1$
Proof
| \(\ds 64\) | \(=\) | \(\ds 2^6\) | ||||||||||||
| \(\ds 81\) | \(=\) | \(\ds 3^4\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 64\) | \(=\) | \(\ds 2^6 \times 3^0\) | |||||||||||
| \(\ds 81\) | \(=\) | \(\ds 2^0 \times 3^4\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \gcd \set {64, 81}\) | \(=\) | \(\ds 2^0 \times 3^0\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1\) |
$\blacksquare$
Sources
- 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {2-4}$ The Fundamental Theorem of Arithmetic: Exercise $6 \ \text{(e)}$