Blackboard Number Operations

CAT 2008 Slot 1 · QA · Easy · Arithmetic

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The integers 1,2,…,401, 2, \dots, 40 are written on a blackboard. The following operation is then repeated 39 times: In each repetition, any two numbers, say aa and bb, currently on the blackboard are erased and a new number a+b−1a + b - 1 is written. What will be the number left on the board at the end?

  1. A.

    820

  2. B.

    821

  3. C.

    781

  4. D.

    819

  5. E.

    780

Answer

C

Explanation

Total sum of the initial numbers =40×412=820= \frac{40 \times 41}{2} = 820.

In each operation, replacing aa and bb with a+b−1a + b - 1 reduces the total sum of the numbers on the board by 1.

After 39 operations, the total sum reduces by 1×39=391 \times 39 = 39.

Therefore, the number left on the board is 820−39=781820 - 39 = 781.

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