## Tuesday, 15 November 2011

### 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)