Why powers of 2?

The sum from 1 to n is equal to $\frac { n(n+1) }{ 2 }$ .

If this sum is divisible by ${ 2 }^{ 10 }$ , then n or (n+1) is divisible by ${ 2 }^{ 11 }$ . Quantity of numbers, such that n is divisible by ${ 2 }^{ 11 }$ , or (n+1) is divisivle by ${ 2 }^{ 11 }$ is equal to $\frac { lcm(2,3,4,...,{ 2 }^{ 100 }) }{ { 2 }^{ 11 } }$ , because $lcm(2,3,4,...,{ 2 }^{ 100 })$ is divisible by ${ 2 }^{ 11 }$ . By adding these two numbers we have $\frac { lcm(2,3,4,...,{ 2 }^{ 100 }) }{ { 2 }^{ 10 } }$ . To find the probability, we need to divide the lenght of whole interval by this quantity. Then ${ P=\frac { \frac { lcm(2,3,4,...,{ 2 }^{ 100 }) }{ { 2 }^{ 10 } } }{ lcm(2,3,4,...,{ 2 }^{ 100 }) } }=\frac { 1 }{ { 2 }^{ 10 } }$ and $\frac { 1 }{ P } ={ 2 }^{ 10 }=1024$ .