The integers 1, 2, …, 40 are written on a blackboard. The
following operation is then repeated 39 times: In each repetition, any two numbers
say a and b, currently on the blackboard are erased and a new number a + b – 1
is written. What will be the number left on the board at the end?

(1) 820 (2) 821 (3) 781 (4) 819 (5) 780

Solution follows here:

__Solution:__
At a time any two numbers can be erased, but let us choose a
pattern:

__Step-1__
1, 40 are erased => 1+40-1 = 40 is the new number written

2, 39 are erased => 2+39-1 = 40 is the new number written

This pattern is repeated until

20, 21 are erased => 20+21-1 = 40 is the new number
written

=> we are left with twenty 40’s on board

__Step-2__
Two 40’s erased and 40+40-1 = 79 is the new number written,
this pattern is repeated

=> we are left with ten 79’s on board

__Step-3__
Two 79’s erased and 79+79-1 = 157 is the new number written,
this pattern is repeated

=> we are left with five 157’s on board

__Step-4__
Two 157’s erased and 157+157-1 = 313 is the new number
written, this pattern is repeated again

=> we are left with two 313’s and one 157 on board

__Step-5__
{(313+313-1)+157}-1 =

**781**the last number left on the board**Answer (3)**
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