Numbers -17 (XAT-2011) (3 Marks)

The micromanometer in a certain factory can measure the pressure inside the gas chamber from 1 unit to 999999 units. Lately this instrument has not been working properly. The problem with the instrument is that it always skips the digit 5 and moves directly from digit 4 to 6. What is the actual pressure inside the gas chamber if the micromanometer displays 003016?

(1)2201          (2)2202          (3)2600          (4)2960          (5)None of these options

Solution follows here:

Solution:

      This can be done by digit-by-digit analysis using concepts of Permutations and Combinations. But the best way out is using number theory as given below:

Here we need to skip number ‘5’ while counting. This means, we need to consider only 9 digits 0, 1, 2, 3, 4, 6, 7, 8, 9 of the base-10 system. ∴The digits to be used in the ‘New system’ are 0, 1, 2, 3, 4, 6, 7, 8, 9. This implies the ‘New system’ is in some-way similar to Base-9 system, which also considers nine single digit numbers (0 to 8). So, we compare the Base-10, Base-9 and the ‘New system’ here:

Base-10	Base-9	New-system		Base-10	Base-9	New-system
0	0	0		9	10	10
1	1	1		10	11	11
2	2	2		11	12	12
3	3	3		12	13	13
4	4	4		13	14	14
5	5	6		14	15	16
6	6	7		15	16	17
7	7	8		16	17	18
8	8	9		17	18	19

      

      Observe that the numbers 0, 1, 2, 3, 4 are same in Base-9 system and the new system. Similarly, numbers 10, 11, 12, 13, 14 are same respectively in Base-9 and the new systems.

But numbers 5,6,7,8 in Base-9 system are respectively equal to 6,7,8,9 in the new system. Similarly, 15,16,17,18 in Base-9 system are respectively equal to 16,17,18,19 in the new system.

      If we go by this logic, 3016 in the new system is equal to 3015 in Base-9 system. Now we need to find (3015)₉ equivalent in Base-10 system.

      (3015)₉ = 5*9⁰ + 1*9¹ + 0*9² + 3*9³ = 5 + 9 + 3*729 = 2201

Answer (1)