Numbers-16 (CAT-2008)

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)