How many terms of the series 1 + 3 + 5 + 7 +... amount to a sum equal to 12345678987654321?

How many terms are identical in the two APs 19, 21, 23,... up to 110 terms and 19, 22, 25, 28,... up to 75 terms?

If $\alpha =\frac{-1+\sqrt{-3}}{2}$ then what is the value of $(1+\alpha ^{19}-\alpha ^{35})^{100}-(1-3\alpha ^{25}+\alpha ^{38})^{50}? $
 

What is the remainder when $5 ^ {99}$ is divided by 13?