Greatest Common Divisor
If m and n are integers, the greatest divisor shared by m and n is the greatest positive integer that divides both m and n.
Notation
We use the expression GCD(a, b, c) to refer to the greatest common divisor of the numbers a, b and c.Properties
- The GCD is always a positive integer.
- The relationship between the GCD and the LCM:
- Consider: PGCD (m, n) = p and PPCM (m, n) = q
- Then: PGCD (m, n) × PPCM (m, n) = m × n
- And we can write: p × q = m × n
- If the GCD (8, 12) = 4 and PPCM (8, 12) = 24, then: 4 × 24 = 8 × 12.
- By extension, we can find the GCD of two or more polynomials. You must factor them first.
- [latex]x^{2}[/latex] – 9 = [latex](x[/latex] + 3)([latex]x[/latex] – 3) [latex]x^{2}[/latex] – [latex]x[/latex] – 12 = ([latex]x[/latex] + 3)([latex]x[/latex] – 4) [latex]x^{2}[/latex] + 6[latex]x[/latex] + 9 = ([latex]x[/latex] + 3)([latex]x[/latex] + 3) Therefore, the GCD of these three polynomials is: ([latex]x[/latex] + 3).