Question

What is the smallest positive integer n such that \sqrt[4]{675 + n} is an integer?

Answer 1

To find the smallest positive integer \( n \) such that \( \sqrt[4]{675 + n} \) is an integer, we set \( \sqrt[4]{675 + n} = k \) where \( k \) is a positive integer. This leads us to the equation:

\[
675 + n = k^4
\]

From this, we can isolate \( n \):

\[
n = k^4 - 675
\]

Since we want \( n \) to be a positive integer, we require:

\[
k^4 > 675
\]

To find the smallest \( k \), we first need to calculate \( \sqrt[4]{675} \). We start by estimating \( \sqrt[4]{675} \):

\[
\sqrt[4]{675} = \sqrt{\sqrt{675}}
\]

Calculating \( \sqrt{675} \), we have:

\[
675 = 25 \times 27 = 25 \times 9 \times 3 = 5^2 \times 3^3
\]

Thus,

\[
\sqrt{675} = \sqrt{25 \times 27} = 5\sqrt{27} = 5 \cdot 3\sqrt{3} = 15\sqrt{3}
\]

Now, estimating \( \sqrt{3} \approx 1.732 \):

\[
\sqrt{675} \approx 15 \times 1.732 \approx 25.98
\]

Now calculating \( \sqrt[4]{675} \):

\[
\sqrt[4]{675} \approx \sqrt{25.98} \approx 5.10
\]

Thus, the smallest integer \( k \) must be at least \( 6 \) because \( 5^4 = 625 \) is less than \( 675 \).

Now we calculate \( n \) for \( k = 6 \):

\[
n = 6^4 - 675 = 1296 - 675 = 621
\]

Next, we check if \( k = 5 \) also provides an integer but we already determined that \( n < 0 \) since \( 5^4 = 625\) which is less than \( 675 \).

Therefore, the minimum \( k \geq 6 \).

To confirm:

\[
k = 6 \Rightarrow n = 621 \Rightarrow 675 + 621 = 1296 \Rightarrow \sqrt[4]{1296} = 6
\]

This confirms that indeed, \( \sqrt[4]{675 + n} \) is an integer.

Thus, the smallest positive integer \( n \) such that \( \sqrt[4]{675 + n} \) is an integer is

\[
\boxed{621}
\]