Euler's theorem states that if two numbers, $a$ and $n$, are coprime (i.e., $\gcd(a, n) = 1$) and $n > 1$, then:

$a^{\phi(n)} \equiv 1 \pmod{n}$

where $\phi(n)$ is Euler's totient function, which counts the number of integers up to $n$ that are coprime to $n$.

Now, let’s check each pair $(a, n)$ to see if $\gcd(a, n) = 1$, which is the condition for Euler's theorem to apply:

a. $(12, 9)$

$\gcd(12, 9) = 3$ Since $\gcd(12, 9) \neq 1$, Euler's theorem does not apply.

b. $(12, 37)$

$\gcd(12, 37) = 1$ Since $\gcd(12, 37) = 1$, Euler's theorem applies.

c. $(12, 14)$

$\gcd(12, 14) = 2$ Since $\gcd(12, 14) \neq 1$, Euler's theorem does not apply.

d. $(15, 12)$

$\gcd(15, 12) = 3$ Since $\gcd(15, 12) \neq 1$, Euler's theorem does not apply.

Conclusion:

Euler's theorem applies only to pair (12, 37).

Would you like more details, or do you have any questions?

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Here are five related questions to further explore this topic:

1. How is Euler's totient function $\phi(n)$ calculated?
2. Can Euler's theorem be applied if $a$ and $n$ are not coprime?
3. What is the significance of the condition $a^{\phi(n)} \equiv 1 \pmod{n}$ in modular arithmetic?
4. How can we use Euler’s theorem to compute large powers modulo $n$?
5. What happens if $\gcd(a, n) \neq 1$ but $a$ is a divisor of $n$?

Tip: Euler’s theorem is particularly useful for simplifying computations of large exponents in modular arithmetic.