91Ó°ÊÓ

Let \(x\) and \(y\) be real numbers. Prove that the greatest integer function satisfies the following properties: (a) \([x+n]=[x]+n\) for any integer \(n\). (b) \([x]+[-x]=0\) or \(-1\), according as \(x\) is an integer or not. \([\) Hint: Write \(x=[x]+\theta\), with \(0 \leq \theta<1\), so that \(-x=-[x]-1+(1-\theta) .]\) (c) \([x]+[y] \leq[x+y]\) and, when \(x\) and \(y\) are positive, \([x][y] \leq[x y]\). (d) \([x / n]=[[x] / n]\) for any positive integer \(n .\) [Hint: Let \(x / n=[x / n]+\theta\), where \(0 \leq \theta<1 ;\) then \([x]=n[x / n]+[n \theta] .]\) (e) \([n m / k] \geq n[m / k]\) for positive integers, \(n, m, k\). (f) \([x]+[y]+[x+y] \leq[2 x]+[2 y]\). [Hint: Let \(x=[x]+\theta, 0 \leq \theta<1\), and \(y=[y]+\theta^{\prime}, 0 \leq \theta^{\prime}<1\). Consider cases in which neither, one, or both of \(\theta\) and \(\theta\) ' are greater than or equal to \(\frac{1}{2}\).]

Find the highest power of 5 dividing \(1000 !\) and the highest power of 7 dividing \(2000 !\).

The Liouville \(\lambda\) -function is defined by \(\lambda(1)=1\) and \(\lambda(n)=(-1)^{k_{1}+k_{2}+\cdots+k_{\prime}}\), if the prime factorization of \(n>1\) is \(n=p_{1}^{k_{1}} p_{2}^{k_{2}}+\cdots p_{r}^{k_{r}} .\) For instance, $$\lambda(360)=\lambda\left(2^{3} \cdot 3^{2} \cdot 5\right)=(-1)^{3+2+1}=(-1)^{6}=1$$ (a) Prove that \(\lambda\) is a multiplicative function. (b) Given a positive integer \(n\), verify that $$ \sum_{d \mid n} \lambda(d)=\left\\{\begin{array}{ll} 1 & \text { if } n=m^{2} \text { for some integer } m \\ 0 & \text { otherwise } \end{array}\right. $$

If \(n\) is a square-free integer, prove that \(\tau(n)=2^{r}\), where \(r\) is the number of prime divisors of \(n\).