5. On Monday, Donut started eating pastries with energy jam. Each day he ate the same number of them and one day he found that only a quarter of his daily portion was left from the initial stock of 340 pastries. On what day of the week did this happen?
# Solution

Let $n$ be the number of days with the same portion, and $s$ be the remainder (a quarter of the portion). Then

$$
2 \cdot 2 \cdot 5 \cdot 17=340=4 s \cdot n+s=s \cdot(4 n+1)
$$

Since the second factor is odd, we get three possible cases.

Either $s=4,4 n+1=85(n=21)$. If 21 days were eaten with a full portion, the shortage was noticed on the 22nd day, i.e., on Monday.

Or $s=20,4 n+1=17(n=4)$. If 4 days were eaten with a full portion, the shortage was noticed on the 5th day, i.e., on Friday.

Or $s=68,4 n+1=5(n=1)$. The condition implicitly suggests that $n>1$, but those who include this third option (on Tuesday) in their answer will also have it counted as correct.

Answer: on Monday or on Friday.