division and numbers

If the number 5a8b7c is completely divisible by 66 and if b + c = 11 and a + b = 14, then find the value of (a – b).

47^th question of MAT 2014 STD V and STD VI

Answer:

The sum of the numbers = 5 + 8 + 7 + a + b + c = 20 + a + b + c   ------------------------- (1)

Substituting (a + b) = 14 in equation 1 we get 

The sum of the numbers = 20 + 14 + c = 34 + c

Now since the number is completely divisible by 66, it must be divisible by 2, 3 and 11.

If the number is divisible by 2 then “c” has to be an even number or 0.

Also, since the number is divisible by 3, hence sum of numbers must be divisible by 3

c	Sum of the numbers = 34 + c	Selection
0	34	3 + 4 = 7 => not divisible by 3
2	36	3 + 6 = 9 => divisible by 3
4	38	3 + 8 = 11 => not divisible by 3
6	40	4 + 0 = 4 => not divisible by 3
8	42	4 + 2 = 6 => divisible by 3

Therefore, c can have only 2 possible values of 2 and 8 to satisfy the divisibility test of 2 and 3.

Now we have to check the divisibility test of 11.

i.e. the sum of the numbers in the even place – the sum of the numbers in the odd place should be a multiple of 11.

Now, sum of the numbers in the even place = 7 + 8 + 5 = 20

Sum of the numbers in the odd place = a + b + c = 14 + c

c	Sum of the numbers in even place  - sum of number in odd place = absolute { 20 – (14 + c) }	Selection
2	4	=> not multiple of  11
8	2	=> not multiple of  11

So, I think no answer is possible with the given set of conditions.