Problem 17

Explanation

We will check each number to see if it satisfies both conditions: divided by $5$ gives a remainder of $3$ AND divided by $4$ gives a remainder of $2$.

Analysis of Statement $1$

Number: $18$.

• Divided by $5$: $18 = 3 \times 5 + 3$. Remainder is $3$. (Satisfied)
• Divided by $4$: $18 = 4 \times 4 + 2$. Remainder is $2$. (Satisfied)

So, the number $18$ is correct.

Analysis of Statement $2$

Number: $38$.

• Divided by $5$: $38 = 7 \times 5 + 3$. Remainder is $3$. (Satisfied)
• Divided by $4$: $38 = 9 \times 4 + 2$. Remainder is $2$. (Satisfied)

So, the number $38$ is correct.

Analysis of Statement $3$

Number: $58$.

• Divided by $5$: $58 = 11 \times 5 + 3$. Remainder is $3$. (Satisfied)
• Divided by $4$: $58 = 14 \times 4 + 2$. Remainder is $2$. (Satisfied)

So, the number $58$ is correct.

Analysis of Statement $4$

Number: $78$.

• Divided by $5$: $78 = 15 \times 5 + 3$. Remainder is $3$. (Satisfied)
• Divided by $4$: $78 = 19 \times 4 + 2$. Remainder is $2$. (Satisfied)

So, the number $78$ is correct.

In conclusion, all four numbers are correct.