Let's not multiply these numbers

12345 is congruent to 8 mod 13, so the remainder is 8 54321 is congruent to 7 mod 13 so the remainder is 7

this means that 12345 is of the form 13k+8, and 54321 is of the form 13j+7 where K and J are integers.

If we were to multiply these numbers, the result would be 13(13kj+8j+7k)+56. we know that the term 13(13kj+8j+7k) is divisible by 13, because it is 13 x the sum of some integers.

Therefore, the remainder is 56, which 4(13)+4. 4, therefore, is our remainder

To get the remainder of two numbers, we can get the remainder of the first number and multiply it by the remainder of the second number and get the remainder of the product

The remainder of $12345$ is $8$ while divided by $13$

The remainder of $54321$ is $7$ while divided by $13$

And the remainder of $7\times8$ is $4$

Therefore, the answer is $\boxed{4}$

In General $\frac{Q}{P}=C \cdot \frac{R}{P}$ if we apply this to our exercise $\frac{12345}{13}= C_1+\frac{R_1}{13}$ $\frac{12345}{13}\cdot 54321= C_1\cdot 54321+\frac{R_1\cdot 54321}{13}$ (1)

$\frac{54321}{13}= C_2 + \frac{R_2}{13}$

$\frac{54321}{13}\cdot R_1= C_2 \cdot R_1+ \frac{R_2\cdot R_1}{13}$ (2)

we substitute (2) in (1)

$\frac{12345}{13}\cdot 54321= C_1\cdot 54321+ C_2 \cdot R_1+ \frac{R_2\cdot R_1}{13}$

Note that OUR remainder is $R_2\cdot R_1$

we find $R_1$ and $R_2$

$\frac{12345}{13}= 949+ \frac{8}{13}$ ; $R_1=8$

$\frac{54321}{13}= 4178+ \frac{7}{13}$ ; $R_2=7$

Therefore our remainder is 56. since it cannot be bigger than 13, we divide again

$\frac{56}{13}= 4 +\frac{4}{13}$

Our remainder is 4

12345*54321 = 670592745, now 670592745/13 then the remainder is 4.

sum the digits firstly,

$1+2+3+4+5=15$ , $5+4+3+2+1=15$ the sum of the digits is same, 15

so , we can write

$15 \times 15$ = $225$ = $\boxed{4} (mod 13)$