An 8 digit number is divisible by 99. If the digits are shuffled, then the number is always divisible by __________.

Given:

An 8 digit number is divisible by 99. If the digits are shuffled, then the number is always divisible by __________.

Concept:

A number is divisible by 99 if and only if it is divisible by both 9 and 11.

Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.

Divisibility by 11: A number is divisible by 11 if the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions is divisible by 11.

Calculation:

Given that the 8-digit number is divisible by 99, it must be divisible by both 9 and 11.

When the digits of a number are shuffled, the sum of its digits remains the same, and so does the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions.

⇒ The sum of the digits remains divisible by 9.

⇒ The difference between the sum of the digits in the odd positions and the sum of the digits in the even positions remains divisible by 11.

Therefore, the number formed by shuffling the digits will also be divisible by both 9 and 11.

∴ The number is always divisible by 9.