Write the number 75 as the sum of several consecutive natural numbers. Find at least four solutions.

(M. Volfová)
If we want to express 75 in the required way using two addends, we are looking for a natural number $x$ such that $75=x+(x+1)=2 x+1$. The only such number is $x=37$, hence

$$
75=37+38 \text {. }
$$

Similarly for three addends, we are looking for a natural solution to the equation $75=x+(x+1)+(x+2)=$ $=3 x+3$, which is $x=24$, hence

$$
75=24+25+26 \text {. }
$$

Using four (similarly eight, twelve, ...) addends, 75 cannot be expressed in this way, because the sum of any four (similarly eight, twelve, ...) consecutive natural numbers is always even.

Five addends correspond to the solution of the equation $75=x+(x+1)+(x+2)+(x+3)+(x+4)=5 x+10$, which is $x=13$, hence

$$
75=13+14+15+16+17
$$

Similarly, we can find the following solutions using six and ten addends, respectively:

$$
\begin{gathered}
75=10+11+12+13+14+15, \\
75=3+4+5+6+7+8+9+10+11+12
\end{gathered}
$$

Note. An alternative notation for three addends might look like this: $75=(y-1)+$ $+y+(y+1)=3 y$, hence $y=25$, which agrees with the previous conclusion. This way of notation is particularly advantageous for odd numbers of addends; for example, 75 cannot be written in the required way using seven or nine addends, because 75 is not divisible by 7 or 9...