Final answer:
To make the number divisible by 8, the possible digits that can be substituted for x are 0 or 4.
Step-by-step explanation:
In order to make the number divisible by 8, we need to find a digit that, when added to the number, will result in a multiple of 8. To do this, we can use the technique of modular arithmetic. If we take the number 678,4x⁶ (where x represents the unknown digit) modulo 8, we can find the remainder when the number is divided by 8. If the remainder is 0, then the number is divisible by 8.
To find the remainder, we first need to express the number as a multiple of 8. We can write 678,4x⁶ as (678,000 + 4x) + 0.0001x⁶, where 0.0001 represents the decimal place of x⁶. Now we can find the remainder using the property in modular arithmetic, which states that (a + b) modulo n is equal to ((a modulo n) + (b modulo n)) modulo n.
By applying this property, we can find that the remainder when (678,000 modulo 8) is 0 and the remainder when (4x modulo 8) is (4x modulo 8). So, in order for the number to be divisible by 8, the remainder when (4x modulo 8) must be 0. The possible values for x that would make the number divisible by 8 are x = 0 or x = 4.