$M=\dfrac{1}{\sqrt[]a-1}-\dfrac{1}{\sqrt[]a}-\dfrac{1-a}{a-\sqrt[]a}$
$ĐKXĐ: x \geq 0; x \neq 0$
$=\dfrac{\sqrt[]a}{\sqrt[]a(\sqrt[]a-1)}-\dfrac{\sqrt[]a-1}{\sqrt[]a(\sqrt[]a-1)}-\dfrac{1-a}{\sqrt[]a(\sqrt[]a-1)}$
$=\dfrac{\sqrt[]a-\sqrt[]a+1-1+a}{\sqrt[]a(\sqrt[]a-1)}$
$=\dfrac{a}{\sqrt[]a(\sqrt[]a-1)}$
$=\dfrac{\sqrt[]a}{\sqrt[]a-1}$
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b, $M∈Z⇔\dfrac{\sqrt[]a}{\sqrt[]a-1}∈Z$
$⇔\sqrt[]a \vdots \sqrt[]a-1$
$⇔\sqrt[]a-1+2 \vdots \sqrt[]a-1$
$⇔2 \vdots \sqrt[]a-1$ (do $\sqrt[]a-1 \vdots \sqrt[]a-1$)
$⇔\sqrt[]a-1∈Ư(2)$
`⇔\sqrt[]a-1∈{-2;-1;1;2}`
`⇔\sqrt[]a∈{-1;0;2;3}`
Mà $\sqrt[]a ≥0∀a$
`⇒\sqrt[]a∈{0;2;3}`
`⇔a∈{0;4;9}(t/m)`
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