Problem 3

Explanation

We are given two conditions for the integer $n$:

1. $n \equiv 2 \pmod{4}$ (remainder is $2$ when divided by $4$)
2. $n \equiv 2 \pmod{6}$ (remainder is $2$ when divided by $6$)

We will check each statement to see if the value of $n$ satisfies both conditions.

Checking Statement 1

Value $n = 18$.

• Divided by $4$: $18 \div 4 = 4$ remainder $2$. (Satisfied)
• Divided by $6$: $18 \div 6 = 3$ remainder $0$. (Not Satisfied)

Thus, statement $(1)$ is incorrect.

Checking Statement 2

Value $n = 24$.

• Divided by $4$: $24 \div 4 = 6$ remainder $0$. (Not Satisfied)

Thus, statement $(2)$ is incorrect.

Checking Statement 3

Value $n = 36$.

• Divided by $4$: $36 \div 4 = 9$ remainder $0$. (Not Satisfied)

Thus, statement $(3)$ is incorrect.

Checking Statement 4

Value $n = 38$.

• Divided by $4$: $38 \div 4 = 9$ remainder $2$. (Satisfied)
• Divided by $6$: $38 \div 6 = 6$ remainder $2$. (Satisfied)

Thus, statement $(4)$ is correct.

Conclusion

Of the four options, only statement $(4)$ satisfies both conditions.