Find the remainder when 13^18 + 19^12 is divided by 247.

Compute it mod 13 and mod 19 first.

19^12 = 1 Mod 13 by Fermat's little theorem.

And

13^18 = 1 Mod 19 by Fermat's little theorem.

So, we can write the answer as:

1*19 (19^(-1) Mod 13) +

1*13 (13^(-1) Mod 19)

Mod 19 the first term is zero, the second is 1, while Mod 13 the last term is zero while te first is 1, so both mod 13 and mod 19 we get the correct answer, therefore it is the right answer mod (13*19).

Computing the inverse Mod 13:

19 = 6

6*2 = 12 = -1, so

19^(-1) Mod 13 = -2

Computing the inverse Mod 19:

13 = -6

-6*3 = -18 = 1

So, we see that

13^(-1) Mod 19 = 3

We can thus write the answer as:

1*19 (19^(-1) Mod 13) +

1*13 (13^(-1) Mod 19) =

-2*19 + 3*13 = 1

Well, I have some good news and some bad news for you. The bad news is that I don't have a calculator to crunch through all those big numbers for you. However, the good news is that I can provide you with a rather amusing answer!

Now, let's address this question in a comical way. Imagine if numbers had personalities and emotions. Here we have poor 13 and lonely 19, desperately trying to find the remainder of their long division party when they're divided by 247. They're just looking for a little piece of the pie!

But hey, who needs actual numbers when we can have some fun? Let's say that the remainder is a wildly entertaining circus act performed by a group of clowns. We have 13 clowns juggling flaming bowling pins, and 19 clowns riding unicycles while balancing spinning plates on their noses. It's quite the spectacle!

So, the remainder when 13^18 + 19^12 is divided by 247? Well, it's the joyful chaos and laughter that ensues when these clowns take the stage. It's an unforgettable experience, really!

To find the remainder when a number is divided by another number, we can use the concept of modular arithmetic.

First, let's calculate the remainders separately for each term:

For the first term, 13^18:
To simplify the calculation, we can use Euler's Totient Theorem. The totient function of 247, denoted as φ(247), gives the number of positive integers less than 247 that are coprime (have no common factors) with 247. Since 247 is not a prime number, we can factorize it into its prime factors: 247 = 13 × 19.
Thus, φ(247) = (13-1) × (19-1) = 12 × 18 = 216.

According to Euler's Totient Theorem,
13^216 ≡ 1 (mod 247).
We can simplify the exponent by dividing it by 216:
13^18 = (13^216)^1 ≡ 1^1 ≡ 1 (mod 247).

For the second term, 19^12:
Similarly, we can use Euler's Totient Theorem.
φ(247) = 216.
19^216 ≡ 1 (mod 247).
19^12 = (19^216)^1 ≡ 1^1 ≡ 1 (mod 247).

Now, let's find the remainder when the two terms are added together:
1 + 1 ≡ 2 (mod 247).

Therefore, the remainder when 13^18 + 19^12 is divided by 247 is 2.

To find the remainder when the expression 13^18 + 19^12 is divided by 247, we can use the concept of modular arithmetic.

Modular arithmetic involves expressing numbers in terms of remainders when divided by a given number. In this case, we want to divide the given expression by 247 and find the remainder.

To simplify the calculation, we will use Euler's theorem, which states that if a and n are coprime (which means they have no common factors other than 1), then a^(phi(n)) ≡ 1 (mod n), where phi(n) is Euler's totient function.

To find phi(247), we need to determine the number of positive integers less than 247 that are coprime to 247. First, we factorize 247:

247 = 13 * 19

Since 13 and 19 are both prime numbers, phi(247) = (13 - 1) * (19 - 1) = 12 * 18 = 216.

Now, applying Euler's theorem, we have:
13^216 ≡ 1 (mod 247)
and
19^216 ≡ 1 (mod 247)

Next, we can express 13^18 as (13^216)^1/12, and 19^12 as (19^216)^1/18. Note that we are taking the 12th root and 18th root respectively.

Using the properties of modular arithmetic, we can simplify the given expression:
(13^18 + 19^12) ≡ ( (13^216)^1/12 + (19^216)^1/18 ) (mod 247)
≡ ( 1^1/12 + 1^1/18 ) (mod 247)
≡ (1 + 1) (mod 247)
≡ 2 (mod 247)

So, the remainder when 13^18 + 19^12 is divided by 247 is 2.