Problem $1$ Find the sum of all natural numbers that make the expression $\sqrt{384 - 12n}$ an integer.

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Q. Problem

1

Find the sum of all natural numbers that make the expression

\sqrt{384 - 12n}

an integer.

Find $n$ for $m^2$ : We need to find natural numbers $n$ such that $\sqrt{384 - 12n}$ is an integer. Let's call the integer $m$ , so we have $m^2 = 384 - 12n$ .
Rearrange to solve for $n$ : Rearrange the equation to solve for $n$ : $n = \frac{384 - m^2}{12}$ .
Determine largest possible $m$ : Since $m^2$ must be less than or equal to $384$ for $n$ to be a natural number, we find the largest possible integer $m$ by taking the square root of $384$ . $m \leq \sqrt{384}$ .
Calculate largest $m$ : Calculate the largest possible integer $m$ : $m \leq \sqrt{384} \approx 19.5959$ . So the largest $m$ is $19$ .
Check values of $m$ : Now we need to find all integer values of $m$ from $1$ to $19$ that make $n$ a natural number when plugged into $n = \frac{384 - m^2}{12}$ .
Calculate $n$ for each $m$ : Check each value of $m$ from $1$ to $19$ and calculate $n$ . If $n$ is a natural number, add it to the sum.
Check for natural number $n$ : For $m = 1$ : $n = (384 - 1^2) / 12 = 383 / 12$ , which is not a natural number.
Identify natural numbers: For $m = 2$ : $n = (384 - 2^2) / 12 = 380 / 12$ , which is not a natural number.
Correct sum calculation: For $m = 3$ : $n = (384 - 3^2) / 12 = 375 / 12$ , which is not a natural number.
Correct sum calculation: For $m = 3$ : $n = (384 - 3^2) / 12 = 375 / 12$ , which is not a natural number.For $m = 4$ : $n = (384 - 4^2) / 12 = 368 / 12$ , which is not a natural number.
Correct sum calculation: For $m = 3$ : $n = (384 - 3^2) / 12 = 375 / 12$ , which is not a natural number.For $m = 4$ : $n = (384 - 4^2) / 12 = 368 / 12$ , which is not a natural number.For $m = 5$ : $n = (384 - 5^2) / 12 = 359 / 12$ , which is not a natural number.
Correct sum calculation: For $m = 3$ : $n = (384 - 3^2) / 12 = 375 / 12$ , which is not a natural number.For $m = 4$ : $n = (384 - 4^2) / 12 = 368 / 12$ , which is not a natural number.For $m = 5$ : $n = (384 - 5^2) / 12 = 359 / 12$ , which is not a natural number.For $m = 6$ : $n = (384 - 6^2) / 12 = 348 / 12 = 29$ , which is a natural number.
Correct sum calculation: For $m = 3$ : $n = (384 - 3^2) / 12 = 375 / 12$ , which is not a natural number.For $m = 4$ : $n = (384 - 4^2) / 12 = 368 / 12$ , which is not a natural number.For $m = 5$ : $n = (384 - 5^2) / 12 = 359 / 12$ , which is not a natural number.For $m = 6$ : $n = (384 - 6^2) / 12 = 348 / 12 = 29$ , which is a natural number.For $m = 7$ : $n = (384 - 7^2) / 12 = 335 / 12$ , which is not a natural number.
Correct sum calculation: For $m = 3$ : $n = (384 - 3^2) / 12 = 375 / 12$ , which is not a natural number.For $m = 4$ : $n = (384 - 4^2) / 12 = 368 / 12$ , which is not a natural number.For $m = 5$ : $n = (384 - 5^2) / 12 = 359 / 12$ , which is not a natural number.For $m = 6$ : $n = (384 - 6^2) / 12 = 348 / 12 = 29$ , which is a natural number.For $m = 7$ : $n = (384 - 7^2) / 12 = 335 / 12$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $0$ : $n = (384 - 3^2) / 12 = 375 / 12$ $1$ , which is not a natural number.
Correct sum calculation: For $m = 3$ : $n = (384 - 3^2) / 12 = 375 / 12$ , which is not a natural number.For $m = 4$ : $n = (384 - 4^2) / 12 = 368 / 12$ , which is not a natural number.For $m = 5$ : $n = (384 - 5^2) / 12 = 359 / 12$ , which is not a natural number.For $m = 6$ : $n = (384 - 6^2) / 12 = 348 / 12 = 29$ , which is a natural number.For $m = 7$ : $n = (384 - 7^2) / 12 = 335 / 12$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $0$ : $n = (384 - 3^2) / 12 = 375 / 12$ $1$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $2$ : $n = (384 - 3^2) / 12 = 375 / 12$ $3$ , which is not a natural number.
Correct sum calculation: For $m = 3$ : $n = (384 - 3^2) / 12 = 375 / 12$ , which is not a natural number.For $m = 4$ : $n = (384 - 4^2) / 12 = 368 / 12$ , which is not a natural number.For $m = 5$ : $n = (384 - 5^2) / 12 = 359 / 12$ , which is not a natural number.For $m = 6$ : $n = (384 - 6^2) / 12 = 348 / 12 = 29$ , which is a natural number.For $m = 7$ : $n = (384 - 7^2) / 12 = 335 / 12$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $0$ : $n = (384 - 3^2) / 12 = 375 / 12$ $1$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $2$ : $n = (384 - 3^2) / 12 = 375 / 12$ $3$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $4$ : $n = (384 - 3^2) / 12 = 375 / 12$ $5$ , which is not a natural number.
Correct sum calculation: For $m = 3$ : $n = (384 - 3^2) / 12 = 375 / 12$ , which is not a natural number.For $m = 4$ : $n = (384 - 4^2) / 12 = 368 / 12$ , which is not a natural number.For $m = 5$ : $n = (384 - 5^2) / 12 = 359 / 12$ , which is not a natural number.For $m = 6$ : $n = (384 - 6^2) / 12 = 348 / 12 = 29$ , which is a natural number.For $m = 7$ : $n = (384 - 7^2) / 12 = 335 / 12$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $0$ : $n = (384 - 3^2) / 12 = 375 / 12$ $1$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $2$ : $n = (384 - 3^2) / 12 = 375 / 12$ $3$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $4$ : $n = (384 - 3^2) / 12 = 375 / 12$ $5$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $6$ : $n = (384 - 3^2) / 12 = 375 / 12$ $7$ , which is not a natural number.
Correct sum calculation: For $m = 3$ : $n = (384 - 3^2) / 12 = 375 / 12$ , which is not a natural number.For $m = 4$ : $n = (384 - 4^2) / 12 = 368 / 12$ , which is not a natural number.For $m = 5$ : $n = (384 - 5^2) / 12 = 359 / 12$ , which is not a natural number.For $m = 6$ : $n = (384 - 6^2) / 12 = 348 / 12 = 29$ , which is a natural number.For $m = 7$ : $n = (384 - 7^2) / 12 = 335 / 12$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $0$ : $n = (384 - 3^2) / 12 = 375 / 12$ $1$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $2$ : $n = (384 - 3^2) / 12 = 375 / 12$ $3$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $4$ : $n = (384 - 3^2) / 12 = 375 / 12$ $5$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $6$ : $n = (384 - 3^2) / 12 = 375 / 12$ $7$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $8$ : $n = (384 - 3^2) / 12 = 375 / 12$ $9$ , which is a natural number.
Correct sum calculation: For $m = 3$ : $n = (384 - 3^2) / 12 = 375 / 12$ , which is not a natural number.For $m = 4$ : $n = (384 - 4^2) / 12 = 368 / 12$ , which is not a natural number.For $m = 5$ : $n = (384 - 5^2) / 12 = 359 / 12$ , which is not a natural number.For $m = 6$ : $n = (384 - 6^2) / 12 = 348 / 12 = 29$ , which is a natural number.For $m = 7$ : $n = (384 - 7^2) / 12 = 335 / 12$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $0$ : $n = (384 - 3^2) / 12 = 375 / 12$ $1$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $2$ : $n = (384 - 3^2) / 12 = 375 / 12$ $3$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $4$ : $n = (384 - 3^2) / 12 = 375 / 12$ $5$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $6$ : $n = (384 - 3^2) / 12 = 375 / 12$ $7$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $8$ : $n = (384 - 3^2) / 12 = 375 / 12$ $9$ , which is a natural number.For $m = 4$ $0$ : $m = 4$ $1$ , which is not a natural number.
Correct sum calculation: For $m = 3$ : $n = (384 - 3^2) / 12 = 375 / 12$ , which is not a natural number.For $m = 4$ : $n = (384 - 4^2) / 12 = 368 / 12$ , which is not a natural number.For $m = 5$ : $n = (384 - 5^2) / 12 = 359 / 12$ , which is not a natural number.For $m = 6$ : $n = (384 - 6^2) / 12 = 348 / 12 = 29$ , which is a natural number.For $m = 7$ : $n = (384 - 7^2) / 12 = 335 / 12$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $0$ : $n = (384 - 3^2) / 12 = 375 / 12$ $1$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $2$ : $n = (384 - 3^2) / 12 = 375 / 12$ $3$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $4$ : $n = (384 - 3^2) / 12 = 375 / 12$ $5$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $6$ : $n = (384 - 3^2) / 12 = 375 / 12$ $7$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $8$ : $n = (384 - 3^2) / 12 = 375 / 12$ $9$ , which is a natural number.For $m = 4$ $0$ : $m = 4$ $1$ , which is not a natural number.For $m = 4$ $2$ : $m = 4$ $3$ , which is not a natural number.
Correct sum calculation: For $m = 3$ : $n = (384 - 3^2) / 12 = 375 / 12$ , which is not a natural number.For $m = 4$ : $n = (384 - 4^2) / 12 = 368 / 12$ , which is not a natural number.For $m = 5$ : $n = (384 - 5^2) / 12 = 359 / 12$ , which is not a natural number.For $m = 6$ : $n = (384 - 6^2) / 12 = 348 / 12 = 29$ , which is a natural number.For $m = 7$ : $n = (384 - 7^2) / 12 = 335 / 12$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $0$ : $n = (384 - 3^2) / 12 = 375 / 12$ $1$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $2$ : $n = (384 - 3^2) / 12 = 375 / 12$ $3$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $4$ : $n = (384 - 3^2) / 12 = 375 / 12$ $5$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $6$ : $n = (384 - 3^2) / 12 = 375 / 12$ $7$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $8$ : $n = (384 - 3^2) / 12 = 375 / 12$ $9$ , which is a natural number.For $m = 4$ $0$ : $m = 4$ $1$ , which is not a natural number.For $m = 4$ $2$ : $m = 4$ $3$ , which is not a natural number.For $m = 4$ $4$ : $m = 4$ $5$ , which is not a natural number.
Correct sum calculation: For $m = 3$ : $n = (384 - 3^2) / 12 = 375 / 12$ , which is not a natural number.For $m = 4$ : $n = (384 - 4^2) / 12 = 368 / 12$ , which is not a natural number.For $m = 5$ : $n = (384 - 5^2) / 12 = 359 / 12$ , which is not a natural number.For $m = 6$ : $n = (384 - 6^2) / 12 = 348 / 12 = 29$ , which is a natural number.For $m = 7$ : $n = (384 - 7^2) / 12 = 335 / 12$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $0$ : $n = (384 - 3^2) / 12 = 375 / 12$ $1$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $2$ : $n = (384 - 3^2) / 12 = 375 / 12$ $3$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $4$ : $n = (384 - 3^2) / 12 = 375 / 12$ $5$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $6$ : $n = (384 - 3^2) / 12 = 375 / 12$ $7$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $8$ : $n = (384 - 3^2) / 12 = 375 / 12$ $9$ , which is a natural number.For $m = 4$ $0$ : $m = 4$ $1$ , which is not a natural number.For $m = 4$ $2$ : $m = 4$ $3$ , which is not a natural number.For $m = 4$ $4$ : $m = 4$ $5$ , which is not a natural number.For $m = 4$ $6$ : $m = 4$ $7$ , which is not a natural number.
Correct sum calculation: For $m = 3$ : $n = (384 - 3^2) / 12 = 375 / 12$ , which is not a natural number.For $m = 4$ : $n = (384 - 4^2) / 12 = 368 / 12$ , which is not a natural number.For $m = 5$ : $n = (384 - 5^2) / 12 = 359 / 12$ , which is not a natural number.For $m = 6$ : $n = (384 - 6^2) / 12 = 348 / 12 = 29$ , which is a natural number.For $m = 7$ : $n = (384 - 7^2) / 12 = 335 / 12$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $0$ : $n = (384 - 3^2) / 12 = 375 / 12$ $1$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $2$ : $n = (384 - 3^2) / 12 = 375 / 12$ $3$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $4$ : $n = (384 - 3^2) / 12 = 375 / 12$ $5$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $6$ : $n = (384 - 3^2) / 12 = 375 / 12$ $7$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $8$ : $n = (384 - 3^2) / 12 = 375 / 12$ $9$ , which is a natural number.For $m = 4$ $0$ : $m = 4$ $1$ , which is not a natural number.For $m = 4$ $2$ : $m = 4$ $3$ , which is not a natural number.For $m = 4$ $4$ : $m = 4$ $5$ , which is not a natural number.For $m = 4$ $6$ : $m = 4$ $7$ , which is not a natural number.For $m = 4$ $8$ : $m = 4$ $9$ , which is not a natural number.
Correct sum calculation: For $m = 3$ : $n = (384 - 3^2) / 12 = 375 / 12$ , which is not a natural number.For $m = 4$ : $n = (384 - 4^2) / 12 = 368 / 12$ , which is not a natural number.For $m = 5$ : $n = (384 - 5^2) / 12 = 359 / 12$ , which is not a natural number.For $m = 6$ : $n = (384 - 6^2) / 12 = 348 / 12 = 29$ , which is a natural number.For $m = 7$ : $n = (384 - 7^2) / 12 = 335 / 12$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $0$ : $n = (384 - 3^2) / 12 = 375 / 12$ $1$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $2$ : $n = (384 - 3^2) / 12 = 375 / 12$ $3$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $4$ : $n = (384 - 3^2) / 12 = 375 / 12$ $5$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $6$ : $n = (384 - 3^2) / 12 = 375 / 12$ $7$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $8$ : $n = (384 - 3^2) / 12 = 375 / 12$ $9$ , which is a natural number.For $m = 4$ $0$ : $m = 4$ $1$ , which is not a natural number.For $m = 4$ $2$ : $m = 4$ $3$ , which is not a natural number.For $m = 4$ $4$ : $m = 4$ $5$ , which is not a natural number.For $m = 4$ $6$ : $m = 4$ $7$ , which is not a natural number.For $m = 4$ $8$ : $m = 4$ $9$ , which is not a natural number.For $n = (384 - 4^2) / 12 = 368 / 12$ $0$ : $n = (384 - 4^2) / 12 = 368 / 12$ $1$ , which is a natural number.
Correct sum calculation: For $m = 3$ : $n = (384 - 3^2) / 12 = 375 / 12$ , which is not a natural number.For $m = 4$ : $n = (384 - 4^2) / 12 = 368 / 12$ , which is not a natural number.For $m = 5$ : $n = (384 - 5^2) / 12 = 359 / 12$ , which is not a natural number.For $m = 6$ : $n = (384 - 6^2) / 12 = 348 / 12 = 29$ , which is a natural number.For $m = 7$ : $n = (384 - 7^2) / 12 = 335 / 12$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $0$ : $n = (384 - 3^2) / 12 = 375 / 12$ $1$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $2$ : $n = (384 - 3^2) / 12 = 375 / 12$ $3$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $4$ : $n = (384 - 3^2) / 12 = 375 / 12$ $5$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $6$ : $n = (384 - 3^2) / 12 = 375 / 12$ $7$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $8$ : $n = (384 - 3^2) / 12 = 375 / 12$ $9$ , which is a natural number.For $m = 4$ $0$ : $m = 4$ $1$ , which is not a natural number.For $m = 4$ $2$ : $m = 4$ $3$ , which is not a natural number.For $m = 4$ $4$ : $m = 4$ $5$ , which is not a natural number.For $m = 4$ $6$ : $m = 4$ $7$ , which is not a natural number.For $m = 4$ $8$ : $m = 4$ $9$ , which is not a natural number.For $n = (384 - 4^2) / 12 = 368 / 12$ $0$ : $n = (384 - 4^2) / 12 = 368 / 12$ $1$ , which is a natural number.For $n = (384 - 4^2) / 12 = 368 / 12$ $2$ : $n = (384 - 4^2) / 12 = 368 / 12$ $3$ , which is not a natural number.
Correct sum calculation: For $m = 3$ : $n = (384 - 3^2) / 12 = 375 / 12$ , which is not a natural number.For $m = 4$ : $n = (384 - 4^2) / 12 = 368 / 12$ , which is not a natural number.For $m = 5$ : $n = (384 - 5^2) / 12 = 359 / 12$ , which is not a natural number.For $m = 6$ : $n = (384 - 6^2) / 12 = 348 / 12 = 29$ , which is a natural number.For $m = 7$ : $n = (384 - 7^2) / 12 = 335 / 12$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $0$ : $n = (384 - 3^2) / 12 = 375 / 12$ $1$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $2$ : $n = (384 - 3^2) / 12 = 375 / 12$ $3$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $4$ : $n = (384 - 3^2) / 12 = 375 / 12$ $5$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $6$ : $n = (384 - 3^2) / 12 = 375 / 12$ $7$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $8$ : $n = (384 - 3^2) / 12 = 375 / 12$ $9$ , which is a natural number.For $m = 4$ $0$ : $m = 4$ $1$ , which is not a natural number.For $m = 4$ $2$ : $m = 4$ $3$ , which is not a natural number.For $m = 4$ $4$ : $m = 4$ $5$ , which is not a natural number.For $m = 4$ $6$ : $m = 4$ $7$ , which is not a natural number.For $m = 4$ $8$ : $m = 4$ $9$ , which is not a natural number.For $n = (384 - 4^2) / 12 = 368 / 12$ $0$ : $n = (384 - 4^2) / 12 = 368 / 12$ $1$ , which is a natural number.For $n = (384 - 4^2) / 12 = 368 / 12$ $2$ : $n = (384 - 4^2) / 12 = 368 / 12$ $3$ , which is not a natural number.The natural numbers that work are $n = (384 - 4^2) / 12 = 368 / 12$ $4$ and $n = (384 - 4^2) / 12 = 368 / 12$ $5$ . Add them up to get the sum.
Correct sum calculation: For $m = 3$ : $n = (384 - 3^2) / 12 = 375 / 12$ , which is not a natural number.For $m = 4$ : $n = (384 - 4^2) / 12 = 368 / 12$ , which is not a natural number.For $m = 5$ : $n = (384 - 5^2) / 12 = 359 / 12$ , which is not a natural number.For $m = 6$ : $n = (384 - 6^2) / 12 = 348 / 12 = 29$ , which is a natural number.For $m = 7$ : $n = (384 - 7^2) / 12 = 335 / 12$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $0$ : $n = (384 - 3^2) / 12 = 375 / 12$ $1$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $2$ : $n = (384 - 3^2) / 12 = 375 / 12$ $3$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $4$ : $n = (384 - 3^2) / 12 = 375 / 12$ $5$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $6$ : $n = (384 - 3^2) / 12 = 375 / 12$ $7$ , which is not a natural number.For $n = (384 - 3^2) / 12 = 375 / 12$ $8$ : $n = (384 - 3^2) / 12 = 375 / 12$ $9$ , which is a natural number.For $m = 4$ $0$ : $m = 4$ $1$ , which is not a natural number.For $m = 4$ $2$ : $m = 4$ $3$ , which is not a natural number.For $m = 4$ $4$ : $m = 4$ $5$ , which is not a natural number.For $m = 4$ $6$ : $m = 4$ $7$ , which is not a natural number.For $m = 4$ $8$ : $m = 4$ $9$ , which is not a natural number.For $n = (384 - 4^2) / 12 = 368 / 12$ $0$ : $n = (384 - 4^2) / 12 = 368 / 12$ $1$ , which is a natural number.For $n = (384 - 4^2) / 12 = 368 / 12$ $2$ : $n = (384 - 4^2) / 12 = 368 / 12$ $3$ , which is not a natural number.The natural numbers that work are $n = (384 - 4^2) / 12 = 368 / 12$ $4$ and $n = (384 - 4^2) / 12 = 368 / 12$ $5$ . Add them up to get the sum. $n = (384 - 4^2) / 12 = 368 / 12$ $6$ . Oops, made a mistake here, let's correct it.

Problem 111 Find the sum of all natural numbers that make the expression 384−12n\sqrt{384 - 12n}384−12n​ an integer.

Problem $1$ Find the sum of all natural numbers that make the expression $\sqrt{384 - 12n}$ an integer.