Question

The sum of first n natural numbers is a three-digit number all of whose digits are the same.What is the value of n ?

Answer 1

In 5 seconds you can solve the equation $\frac{n(n+1)}{2}=aaa$ (111, 222, etc)

$\frac{n(n+1)}{2}=aaa=a\times 111=a\times 3\times 37$ $\Rightarrow n(n+1)=6ax37$

The L.H.S of the equation n(n + 1) is a product of two consecutive
natural numbers Therefore R.H.S. should also be a product of two
consecutive natural numbers One of the numbers is 37 Therefore the other number, 6a consecutive to 37 can only be 36 giving a = 6 and n = 36

Therefore 36 numbers have been summed up and their sum
is equal to 666