Distribution of decimals in some sequences

A classic problem is of showing the existing of perfect squares or powers of 2 which begin with a combination of digits,for example:
Prove that there $\exists n \in N$ so the first four digits of $2^n$be 2002.
Proof:The conclusion is equivalent to this one:$2002 \cdot 10^k \leq 2^n \leq 2003\cdot 10^k \Leftrightarrow k+lg2002\leq nlg2 \leq k+lg2003 \Leftrightarrow lg 2002 \leq nlg 2-k \leq lg 2003$
Because $lg2 \notin Q$ from Kronecker's theorem the set $\{nlg2-k|n,k \in N\}$is dense in R so it has elements in $[lg2002,lg2003)$.

Exercise:
Prove that $\exists m,n \in Z$so that the first three decimals of $m\sqrt{2}+n\sqrt{3}$ are 101