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Quiz 18
1.) Find the least common multiple of the numbers $3$, $6$, and $5$.
Solution: Since $3$ is prime, $6=3 \cdot 2$, and $5$ is prime, the least common multiple is
$$2 \cdot 3 \cdot 5 = 30.$$
2.) Find the least common multiple of $(x+1)(x+2)^2$, $(x+1)^2(x+2)$, and $(x+1)(x+2)$.
Solution: The least common multiple is
$$(x+1)^2(x+2)^2.$$
3.) Find the least common multiple of $x^2+4x+4$, $x^2+3x+2$, and $x^2+5x+6$.
Solution: Since we can factor
$$x^2+4x+4=(x+2)^2,$$
$$x^2+3x+2=(x+1)(x+2),$$
and
$$x^2+5x+6=(x+3)(x+2),$$
the least common multiple is
$$(x+1)(x+2)^2(x+3).$$
4.) Add and simplify:
$$\dfrac{1}{2} + \dfrac{1}{3}.$$
Solution: The least common denominator is $6$. Therefore $\dfrac{1}{2}=\dfrac{3}{6}$ and $\dfrac{1}{3}=\dfrac{2}{6}$. Now we may add:
$$\dfrac{1}{2} +\dfrac{1}{3} = \dfrac{3}{6} + \dfrac{2}{6} = \dfrac{1+2}{6} = \dfrac{5}{6}.$$
5.) Add and simplify:
$$\dfrac{1}{x+1} + \dfrac{1}{x+2}.$$
Solution: The least common denominator is $(x+1)(x+2)$, so we may rewrite
$$\dfrac{1}{x+1} = \dfrac{x+2}{(x+1)(x+2)},$$
and
$$\dfrac{1}{x+2} = \dfrac{x+1}{(x+1)(x+2)}.$$
Now we may add:
$$\dfrac{1}{x+1} + \dfrac{1}{x+2} = \dfrac{x+2}{(x+1)(x+2)} + \dfrac{x+1}{(x+1)(x+2)} = \dfrac{2x+3}{(x+1)(x+2)}.$$
