GCD from Prime Decomposition/Examples/39, 102 and 75
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Example of Use of GCD from Prime Decomposition
The greatest common divisor of $39$, $102$ and $75$ is:
- $\gcd \set {39, 102, 75} = 3$
Proof
\(\ds 39\) | \(=\) | \(\ds 3 \times 13\) | ||||||||||||
\(\ds 102\) | \(=\) | \(\ds 2 \times 3 \times 17\) | ||||||||||||
\(\ds 75\) | \(=\) | \(\ds 3 \times 5^2\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds 39\) | \(=\) | \(\ds 2^0 \times 3^1 \times 5^0 \times 13^1 \times 17^0\) | |||||||||||
\(\ds 102\) | \(=\) | \(\ds 2^1 \times 3^1 \times 5^0 \times 13^0 \times 17^1\) | ||||||||||||
\(\ds 75\) | \(=\) | \(\ds 2^0 \times 3^1 \times 5^2 \times 13^0 \times 17^0\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \gcd \set {39, 102, 75}\) | \(=\) | \(\ds 2^0 \times 3^1 \times 5^0 \times 13^0 \times 17^0\) | |||||||||||
\(\ds \) | \(=\) | \(\ds 3\) |
$\blacksquare$
Sources
- 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {2-4}$ The Fundamental Theorem of Arithmetic: Exercise $11$