Question 140
Caboyara, famous Australian circus performer, puts on a great trick this year too. He sets up a spectacular staircase with $N = p_{1} \cdot p_{2} \cdot \ldots \cdot p_{2015}$ steps, where $p_{1}, p_{2}, \ldots, p_{2015}$ are distinct primes; the steps corresponding to divisors of $N$ (including the first and the $N$-th step) are special and are initially lit green.
During the show, 2015 trained kangaroos climb the staircase one after another; for $i = 1, 2, \ldots, 2015$, the $i$-th kangaroo jumps $p_{i}$ steps at a time, starting at the foot of the staircase (it jumps onto step $p_{i}$, then onto $2 p_{i}$, and so on until it reaches step $N$). Every time a kangaroo jumps onto a special step, that step changes colour: from green it becomes red, from red green.
How many special steps will be lit green at the end of the show?
(A) $2^{2015} - 2^{1008}$
(B) $2^{2014}$
(C) $2^{2014} - 2^{1007}$
(D) $2^{2013}$
(E) $2015 \cdot 2^{1008}$
During the show, 2015 trained kangaroos climb the staircase one after another; for $i = 1, 2, \ldots, 2015$, the $i$-th kangaroo jumps $p_{i}$ steps at a time, starting at the foot of the staircase (it jumps onto step $p_{i}$, then onto $2 p_{i}$, and so on until it reaches step $N$). Every time a kangaroo jumps onto a special step, that step changes colour: from green it becomes red, from red green.
How many special steps will be lit green at the end of the show?
(A) $2^{2015} - 2^{1008}$
(B) $2^{2014}$
(C) $2^{2014} - 2^{1007}$
(D) $2^{2013}$
(E) $2015 \cdot 2^{1008}$