Question

A number 57a2b is divisible by 9 and 36b2 is divisible by 11, where a and b are the
missing digits. Which of these could be the relation between a and b?
Option 1: a= 2b
Option 2: b − a = 3
Option 3: ab
Option 4: a – b = 3

Answer 1

Given:

A number 57a2b is divisible by 9 and 36b2 is divisible by 11, where a and b are missing digits.

To Find:

The value of a and b.

Solution:

The given question can be solved using the divisibility concepts.

1. It is given that the number 57a2b is divisible by 9. For a number to be divisible by 9 the sum of the digits must be divisible by 9, Therefore,

=> 5 + 7 + a + 2 +b is divisible by 9,

=> 14 + a + b is divisible by 9.

2. It is mentioned that 36b2 is divisible by 11. For a number to be divisible by 11, the difference of the sum of the digits in the even places with the sum of the digits in the odd places must be a multiple of 11. Therefore,

=> (3 + b) - (8) is divisible by 11,

=> b - 5 is divisible by 11,

=> b = 5.

3. Substitute the value of b as 5 in the first point. We get,

=> 14 + 5 + a is divisible by 9,

=> 19 + a is divisible by 9,

=> a =8 satisfies the divisibility.

=> a=8.

Therefore, the value of a and b is 5 and 8 respectively.  

Answer 2

a - b = 3  if A number 57a2b is divisible by 9 and 36b2 is divisible by 11, where a and b are the missing digits

Given:

• A number 57a2b is divisible by 9
• A number 36b2 is divisible by 11
• a and b are missing digits

To Find:

• Relation between a and b from below options
• Option 1: a= 2b
• Option 2: b − a = 3
• Option 3: ab
• Option 4: a – b = 3

Solution:

Step 1:

A number is divisible by 9 if sum of digits is divisible by 9

57a2b

sum of digits = 5 + 7 + a + 2 + b

= 14 + a + b

a + b must be 4   or 13

as 14 + 4 = 18  and 14 + 13 = 27 is divisible by 9

a + b = 4

a + b = 13

Step 2:

A number is divisible by 11 if difference between digits at odd places and even places is divisible by 11

36b2

(3 + b) - (6 + 2)

= b - 5

b must be 5

as 5 - 5 = 0 is divisible by 11

Step 3:

a + b = 4 is not possible as b = 5

Hence a + b = 13

=> a + 5 = 13

=> a = 8

a = 8  and b = 5

=> a - b = 8 - 5 = 3

a - b = 3 is correct option

Option 4) a - b = 3