Can You Tell the Seed Number

First, the seed number of a number, $n$ is equal to $n \pmod{9}$ . This is because the seed number is found by repeatedly calculating the digital root from a number, until the result is a one-digit number. The modulus of the digital root of a number is always the same as the modulus of the number.

Since $2^{6} \equiv 1 \pmod{9}$ , we have

$2^{56789} \equiv (2^{6})^{9464} \times 2^{5} \equiv 32 \equiv 5 \pmod{9}$ .

Hence, the seed number of $2^{56789}$ is $\boxed{5}$ .

By considering the pattern of the seed numbers of the sequence we can easily find the solution like this,

$2^{1}=2$

$2^{2}=4$

$2^{3}=8$

$2^{4}=16\rightarrow1+6=7$

$2^{5}=32\rightarrow 3+2=5$

$2^{6}=64\rightarrow 6+4=10\rightarrow1+0=1$

$2^{7}=128\rightarrow 1+2+8=11\rightarrow1+1=2$

$2^{8}=256\rightarrow 2+5+6=13\rightarrow1+3=4$

$2^{9}=512\rightarrow 5+1+2=8$

$2^{10}=1024\rightarrow 1+0+2+4=7$ .................................

So we can observe a pattern in the sum of the digits that is going on and on,

$2,4,8,7,5,1$

$2,4,8,7,5,1........$

The pattern consists of 6 values, so to check the seed number we must divide 56789 by 6.

Then you will obtain a remainder of $5$ .

So the 5th number in our pattern which is $5$ is the correct answer.

1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,............... 1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768,65536,131072,......