For which of the following pairs $
(
a
,
n
)
$ does Euler's theorem apply

Question 1Answer

a.
(12,9)


b.
(12,37)


c.
(12,14)


d.
(15,12)

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.

For which of the following pairs (a, n) does Euler's theorem apply?
a. (12,9)
b. (12,37)
c. (12,14)
d. (15,12)

Find out for which pairs (a, n) Euler's theorem applies by analyzing their GCD values and applying modular arithmetic. This problem introduces core concepts from number theory and is ideal for undergraduate students studying cryptography.

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