- #1
anemone
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Let $a$ and $b$ be two different primes. Prove that
$\displaystyle\left\lfloor\dfrac{a}{b} \right\rfloor+\left\lfloor\dfrac{2a}{b} \right\rfloor+\left\lfloor\dfrac{3a}{b} \right\rfloor+\cdots+\left\lfloor\dfrac{(b-1)a}{b} \right\rfloor=\dfrac{(a-1)(b-1)}{2}$.
$\displaystyle\left\lfloor\dfrac{a}{b} \right\rfloor+\left\lfloor\dfrac{2a}{b} \right\rfloor+\left\lfloor\dfrac{3a}{b} \right\rfloor+\cdots+\left\lfloor\dfrac{(b-1)a}{b} \right\rfloor=\dfrac{(a-1)(b-1)}{2}$.