Question
You are given the numbers, (30 + n, 1/5, \sqrt{n +144}. Find the smallest value of n so that all of the numbers in the set are natural numbers.
Answers
GPT-4o mini
To determine the smallest value of \( n \) such that all numbers in the set \( (30+n, \frac{1}{5}, \sqrt{n + 144}) \) are natural numbers, we need to analyze each number in the set:
1. **For the number \( 30 + n \) to be a natural number:**
Since \( n \) must be a natural number, this will naturally be satisfied as long as \( n \) is a natural number.
2. **For the number \( \frac{1}{5} \) to be a natural number:**
\( \frac{1}{5} \) is not a natural number. Natural numbers are defined as positive integers (1, 2, 3, ...), and \( \frac{1}{5} \) does not qualify. Thus, there's likely a misunderstanding here; it implies that \( \frac{1}{5} \) is not part of the relevant numbers we need to analyze further. We will focus only on \( 30+n \) and \( \sqrt{n+144} \).
3. **For the number \( \sqrt{n + 144} \) to be a natural number:**
This means that \( n + 144 \) must be a perfect square. Let’s express this as:
\[
n + 144 = k^2
\]
for some natural number \( k \). Rearranging gives:
\[
n = k^2 - 144
\]
Since \( n \) must also be non-negative (as it is a natural number), we have:
\[
k^2 - 144 \geq 0
\]
which simplifies to:
\[
k^2 \geq 144
\]
Thus, \( k \) must be at least 12, since \( 12^2 = 144 \).
Now, we can check values of \( k \) starting from 12:
- When \( k = 12 \):
\[
n = 12^2 - 144 = 144 - 144 = 0 \quad (\text{Not a natural number})
\]
- When \( k = 13 \):
\[
n = 13^2 - 144 = 169 - 144 = 25 \quad (\text{This is a natural number})
\]
- When \( k = 14 \):
\[
n = 14^2 - 144 = 196 - 144 = 52 \quad (\text{This is also a natural number})
\]
It is clear from above that the smallest suitable \( n \) occurs at \( k = 13 \), producing:
- \( n = 25 \)
Now we will check our results:
1. **Check \( 30 + n \):**
\[
30 + 25 = 55 \quad \text{(Natural number)}
\]
2. **Check \( \sqrt{n + 144} \):**
\[
\sqrt{25 + 144} = \sqrt{169} = 13 \quad \text{(Natural number)}
\]
Hence, both conditions are satisfied when \( n = 25 \).
Thus, the smallest value of \( n \) so that all of the numbers in the set are natural numbers is:
\[
\boxed{25}
\]
1. **For the number \( 30 + n \) to be a natural number:**
Since \( n \) must be a natural number, this will naturally be satisfied as long as \( n \) is a natural number.
2. **For the number \( \frac{1}{5} \) to be a natural number:**
\( \frac{1}{5} \) is not a natural number. Natural numbers are defined as positive integers (1, 2, 3, ...), and \( \frac{1}{5} \) does not qualify. Thus, there's likely a misunderstanding here; it implies that \( \frac{1}{5} \) is not part of the relevant numbers we need to analyze further. We will focus only on \( 30+n \) and \( \sqrt{n+144} \).
3. **For the number \( \sqrt{n + 144} \) to be a natural number:**
This means that \( n + 144 \) must be a perfect square. Let’s express this as:
\[
n + 144 = k^2
\]
for some natural number \( k \). Rearranging gives:
\[
n = k^2 - 144
\]
Since \( n \) must also be non-negative (as it is a natural number), we have:
\[
k^2 - 144 \geq 0
\]
which simplifies to:
\[
k^2 \geq 144
\]
Thus, \( k \) must be at least 12, since \( 12^2 = 144 \).
Now, we can check values of \( k \) starting from 12:
- When \( k = 12 \):
\[
n = 12^2 - 144 = 144 - 144 = 0 \quad (\text{Not a natural number})
\]
- When \( k = 13 \):
\[
n = 13^2 - 144 = 169 - 144 = 25 \quad (\text{This is a natural number})
\]
- When \( k = 14 \):
\[
n = 14^2 - 144 = 196 - 144 = 52 \quad (\text{This is also a natural number})
\]
It is clear from above that the smallest suitable \( n \) occurs at \( k = 13 \), producing:
- \( n = 25 \)
Now we will check our results:
1. **Check \( 30 + n \):**
\[
30 + 25 = 55 \quad \text{(Natural number)}
\]
2. **Check \( \sqrt{n + 144} \):**
\[
\sqrt{25 + 144} = \sqrt{169} = 13 \quad \text{(Natural number)}
\]
Hence, both conditions are satisfied when \( n = 25 \).
Thus, the smallest value of \( n \) so that all of the numbers in the set are natural numbers is:
\[
\boxed{25}
\]