Problem:

What is the value of x so that the seven digit number 8439 x 53 is divisible by 99?

Solution:

To find the value of x, we need to understand the properties of 99. A number is divisible by 99 if it is divisible by both 9 and 11.

Let's first check the divisibility by 9. The sum of the digits of the number 8439 x 53 is:
8 + 4 + 3 + 9 + 5 + 3 = 32 + x
For the number to be divisible by 9, the sum of its digits must be divisible by 9. Therefore, we need to find the value of x that makes (32 + x) divisible by 9.

If we divide (32 + x) by 9, the remainder must be zero. We can write this as:
32 + x ≡ 0 (mod 9)
Solving for x, we get:
x ≡ -32 ≡ -5 (mod 9)

Now, let's check the divisibility by 11. To do this, we need to find the alternating sum of the digits of the number:
8 - 4 + 3 - 9 + x - 5 + 3 = x - 4
For the number to be divisible by 11, the alternating sum of its digits must be divisible by 11. Therefore, we need to find the value of x that makes (x - 4) divisible by 11.

If we divide (x - 4) by 11, the remainder must be zero. We can write this as:
x - 4 ≡ 0 (mod 11)
Solving for x, we get:
x ≡ 4 (mod 11)

Now we have two congruences for x: x ≡ -5 (mod 9) and x ≡ 4 (mod 11). To find the value of x that satisfies both congruences, we can use the Chinese Remainder Theorem.

We can write the first congruence as x = 9k - 5 for some integer k, and substitute this into the second congruence:
9k - 5 ≡ 4 (mod 11)
Simplifying, we get:
9k ≡ 9 (mod 11)
Multiplying both sides by the inverse of 9 modulo 11 (which is 5), we get:
k ≡ 5 (mod 11)

Substituting this into x = 9k - 5, we get:
x = 9(5) - 5 = 40

Therefore, the value of x that makes the seven digit number 8439 x 53 divisible by 99 is 4.

Answer:

Option (b) 4 is the correct answer.