Answer :
To determine which tens digit to choose for 226 so that the resulting number is divisible by 6, we need to understand the rules of divisibility for 6. A number is divisible by 6 if and only if it is divisible by both 2 and 3.
### Step-by-Step Solution:
1. Given Number Structure:
The number given is [tex]\(226\)[/tex]. We need to replace the ‘tens’ digit (which is currently 2 in the number 226) with one of the options (0, 1, 3, or 2) and check for divisibility.
2. Check Divisibility by 2:
A number is divisible by 2 if its last digit is even. For any choice of tens digit, our number will always end in 6 (the last digit), which is even. Thus, any constructed number will be divisible by 2.
3. Check Divisibility by 3:
A number is divisible by 3 if the sum of its digits is divisible by 3.
Let's calculate the sum of the digits for each choice:
[tex]\[ \text{New number} = 200 + (\text{chosen tens digit} \times 10) + 6 \][/tex]
- For [tex]\( \text{chosen tens digit} = 0 \)[/tex]:
[tex]\[ \text{Number} = 206 \\ \text{Sum of digits} = 2 + 0 + 6 = 8 \quad (\text{not divisible by 3}) \][/tex]
- For [tex]\( \text{chosen tens digit} = 1 \)[/tex]:
[tex]\[ \text{Number} = 216 \\ \text{Sum of digits} = 2 + 1 + 6 = 9 \quad (\text{divisible by 3}) \][/tex]
- For [tex]\( \text{chosen tens digit} = 2 \)[/tex]:
[tex]\[ \text{Number} = 226 \\ \text{Sum of digits} = 2 + 2 + 6 = 10 \quad (\text{not divisible by 3}) \][/tex]
- For [tex]\( \text{chosen tens digit} = 3 \)[/tex]:
[tex]\[ \text{Number} = 236 \\ \text{Sum of digits} = 2 + 3 + 6 = 11 \quad (\text{not divisible by 3}) \][/tex]
4. Conclusion:
From the sums calculated, only the number 216 (which uses the tens digit of 1) is divisible by both 2 and 3, making it also divisible by 6.
Therefore, the correct choice is:
[tex]\[ \boxed{1} \][/tex]
So the best answer from the choices provided is:
[tex]\[ \boxed{\text{B}} \][/tex]
### Step-by-Step Solution:
1. Given Number Structure:
The number given is [tex]\(226\)[/tex]. We need to replace the ‘tens’ digit (which is currently 2 in the number 226) with one of the options (0, 1, 3, or 2) and check for divisibility.
2. Check Divisibility by 2:
A number is divisible by 2 if its last digit is even. For any choice of tens digit, our number will always end in 6 (the last digit), which is even. Thus, any constructed number will be divisible by 2.
3. Check Divisibility by 3:
A number is divisible by 3 if the sum of its digits is divisible by 3.
Let's calculate the sum of the digits for each choice:
[tex]\[ \text{New number} = 200 + (\text{chosen tens digit} \times 10) + 6 \][/tex]
- For [tex]\( \text{chosen tens digit} = 0 \)[/tex]:
[tex]\[ \text{Number} = 206 \\ \text{Sum of digits} = 2 + 0 + 6 = 8 \quad (\text{not divisible by 3}) \][/tex]
- For [tex]\( \text{chosen tens digit} = 1 \)[/tex]:
[tex]\[ \text{Number} = 216 \\ \text{Sum of digits} = 2 + 1 + 6 = 9 \quad (\text{divisible by 3}) \][/tex]
- For [tex]\( \text{chosen tens digit} = 2 \)[/tex]:
[tex]\[ \text{Number} = 226 \\ \text{Sum of digits} = 2 + 2 + 6 = 10 \quad (\text{not divisible by 3}) \][/tex]
- For [tex]\( \text{chosen tens digit} = 3 \)[/tex]:
[tex]\[ \text{Number} = 236 \\ \text{Sum of digits} = 2 + 3 + 6 = 11 \quad (\text{not divisible by 3}) \][/tex]
4. Conclusion:
From the sums calculated, only the number 216 (which uses the tens digit of 1) is divisible by both 2 and 3, making it also divisible by 6.
Therefore, the correct choice is:
[tex]\[ \boxed{1} \][/tex]
So the best answer from the choices provided is:
[tex]\[ \boxed{\text{B}} \][/tex]
