To determine the number of possible digits for [tex]\( x \)[/tex] such that the number [tex]\( 10x78x \)[/tex] is divisible by 3, we need to use the rule for divisibility by 3. According to this rule, a number is divisible by 3 if and only if the sum of its digits is divisible by 3.
Let's analyze the number [tex]\( 10x78x \)[/tex]:
1. The digits in the number are 1, 0, [tex]\( x \)[/tex], 7, 8, and [tex]\( x \)[/tex].
2. Summing these digits, we get:
[tex]\[
1 + 0 + x + 7 + 8 + x
\][/tex]
Simplifying the sum, we obtain:
[tex]\[
1 + 0 + 7 + 8 + 2x = 16 + 2x
\][/tex]
For [tex]\( 10x78x \)[/tex] to be divisible by 3, the sum [tex]\( 16 + 2x \)[/tex] must be divisible by 3.
We need to check each possible digit [tex]\( x \)[/tex] (from 0 to 9) to see which values make [tex]\( 16 + 2x \)[/tex] divisible by 3.
Let's go through each digit systematically:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[
16 + 2 \cdot 0 = 16 \quad (\text{not divisible by 3})
\][/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\[
16 + 2 \cdot 1 = 18 \quad (\text{divisible by 3})
\][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[
16 + 2 \cdot 2 = 20 \quad (\text{not divisible by 3})
\][/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[
16 + 2 \cdot 3 = 22 \quad (\text{not divisible by 3})
\][/tex]
- For [tex]\( x = 4 \)[/tex]:
[tex]\[
16 + 2 \cdot 4 = 24 \quad (\text{divisible by 3})
\][/tex]
- For [tex]\( x = 5 \)[/tex]:
[tex]\[
16 + 2 \cdot 5 = 26 \quad (\text{not divisible by 3})
\][/tex]
- For [tex]\( x = 6 \)[/tex]:
[tex]\[
16 + 2 \cdot 6 = 28 \quad (\text{not divisible by 3})
\][/tex]
- For [tex]\( x = 7 \)[/tex]:
[tex]\[
16 + 2 \cdot 7 = 30 \quad (\text{divisible by 3})
\][/tex]
- For [tex]\( x = 8 \)[/tex]:
[tex]\[
16 + 2 \cdot 8 = 32 \quad (\text{not divisible by 3})
\][/tex]
- For [tex]\( x = 9 \)[/tex]:
[tex]\[
16 + 2 \cdot 9 = 34 \quad (\text{not divisible by 3})
\][/tex]
From this evaluation, we see that the values of [tex]\( x \)[/tex] that make [tex]\( 16 + 2x \)[/tex] divisible by 3 are [tex]\( x = 1 \)[/tex], [tex]\( x = 4 \)[/tex], and [tex]\( x = 7 \)[/tex].
Thus, the number of possible digits for [tex]\( x \)[/tex] such that [tex]\( 10x78x \)[/tex] is divisible by 3 is [tex]\( \boxed{3} \)[/tex].
