1. Marco thought of a six-digit number, whose digit in the ten-thousands place is 5. Then, he moved the digit 5 to the units place, the order of the other digits remained unchanged, and he got a number that is four times smaller than the thought number. What number did Marco think of?
Solution. From the condition of the problem it follows:

$\overline{5 a b c d e}=4 \cdot \overline{a b c d e 5}$

$500000+10000 a+1000 b+100 c+10 d+e=4(100000 a+10000 b+1000 c+100 d+10 e+5)$

$500000-20+(10000-400000) a+(1000-40000) b+(10-4000) c+(10-400) d+(1-40) d=0$
$499980-39(10000 a+1000 b+100 c+10 d+e)=0$

$39 \cdot \overline{a b c d e}=499980$

$\overline{a b c d e}=12820$

Finally, the required number is $\overline{5 a b c d e}=512820$.