Annihilation

It's not hard to see that the rule for particles dividing is the same as the rule for generating binomial coefficients modulo $2.$ (Odd = particle, even = no particle; odd + even makes a particle, even + even makes no particle, odd + odd means the particles annihilate, so no particle.)

So the number of particles after $t$ seconds is the number of odd entries in the $t$ th row of Pascal's triangle. When $t=2^{145} + 2,$ there are exactly $\fbox{4}$ such entries, by Lucas' theorem .

Positions of particles at time $t=2^n$ are $(-2^n,2^n)$ . Since for large $n$ the separation of the particles is much larger than $1$ , the positions of particles at time $t=2^n+2$ are $(-2^n-2,-2^n+2,2^n-2,2^n+2)$ . So there will be $\boxed 4$ particles in total at that time.