Antônio has a parrot that performs fantastic calculations with integers. When Antônio whispers certain numbers into its ear, the parrot multiplies that number by 5, then adds 14, divides the result by 6, and finally subtracts 1, shouting the result afterward. However, the parrot knows nothing about decimals, so sometimes it cannot shout out an answer.

(a) If Antônio whispers the number 8, what number will the parrot shout?

(b) If the parrot shouted 3, what number did Antônio whisper into its ear?

(c) Why does the parrot never shout the number 7?

(d) Which numbers, when whispered by Antônio, elicit a response from the parrot?
(a) We have $8 \xrightarrow{\times 5} 40 \xrightarrow{+14} 54 \xrightarrow{\dot{\circ} 6} 9 \xrightarrow{-1} 8$. Therefore, the parrot shouts 8.

(b) We should perform the inverse operation of what the parrot did, starting from the last operation, that is, add 1 to the number, multiply the number by 6, then subtract 14, and finally divide the result by 5:

$$
3 \xrightarrow{+1} 4 \xrightarrow{\times 6} 24 \xrightarrow{-14} 10 \xrightarrow{\div 5} 2.
$$

Therefore, Antônio whispered 2 into the parrot's ear.

(c) Observe that $7 \xrightarrow{+1} 8 \xrightarrow{\times 6} 48 \xrightarrow{-14} 34 \xrightarrow{\dot{* 5}} 6.8$. Since 6.8 is not an integer, Antônio will not whisper it into the parrot's ear, and even if he did, the parrot would not know how to perform the first operation, which would be to multiply $6.8 \times 5$.

(d) When Antônio whispers a number $n$, the parrot performs the operations

$$
n \xrightarrow{\times 5} 5 n \xrightarrow{+14} 5 n+14 \xrightarrow{\dot{\circ}} \frac{5 n+14}{6} \xrightarrow{-1} \frac{5 n+14}{6}-1.
$$

The parrot will only know how to calculate the answer if $5 n+14$ is divisible by 6, that is, if it is of the form $6 k$, with $k$ a non-negative integer. If $5 n+14=6 k$, then $5 n+2=6(k-2)$ and, multiplying both sides by 5, we get $25 n+10=6(5 k-10)$, which simplifies to $n+24 n=25 n=6(5 k-10)-12+2$, or $n=6(5 k-12)+2-24 n=6(5 k-12-4 n)+2$. Thus, if Antônio whispers a number $n$ of the form $6 m+2$, the parrot performs the operations

$$
6 m+2 \xrightarrow{\times 5} 30 m+10 \xrightarrow{+14} 30 m+24 \xrightarrow{\dot{\circ} 6} 5 m+4 \xrightarrow{-1} 5 m+3
$$

and shouts the number $5 m+3$. If $n$ is not of this form, the parrot remains silent. Therefore, Antônio can only whisper the numbers

$$
2,8,14,20,26,32,38, \ldots
$$

and the parrot can only respond, respectively,

$$
3,8,13,18,23,28,33, \ldots
$$