Given: A B 45896 = C D and C D = 96023 B

To find: Value of A

Let's start by breaking down the given equation:

A B 45896 = C D
=> AB × 10000 + 45896 = CD (expanded form)

C D = 96023 B
=> CD = 96000 + 23B

Substituting the value of CD from the second equation into the first equation, we get:

AB × 10000 + 45896 = 96000 + 23B

Simplifying this equation, we get:

10000AB - 23B = 50104

We need to find the value of A. To do that, we need to express B in terms of A:

10000AB - 23B = 50104
=> 23B = 10000AB - 50104
=> B = (10000AB - 50104)/23

Substituting this value of B into the equation CD = 96023 B, we get:

CD = 96023 × (10000AB - 50104)/23

Simplifying this equation, we get:

CD = (96000/23)AB - (2087 × 23)

Since CD is an integer, (96000/23)AB must be a multiple of 23. Therefore, AB must be a multiple of 23/96000.

AB can be expressed as:

AB = (50104 + 23B)/10000

Substituting the value of B, we get:

AB = (50104 + 23[(10000AB - 50104)/23])/10000
=> AB = 50127/10000 × AB - 2087/10000

Simplifying this equation, we get:

AB - (50127/10000)AB = 2087/10000
=> (49973/10000)AB = 2087/10000
=> AB = 2087/49973

Since AB is an integer, the numerator of this fraction must be a multiple of 49973. Therefore, A must be a factor of 2087.

The factors of 2087 are 1 and 2087. Therefore, the only possible value of A is 2087.

Hence, the answer is option B (50127).

50127