Understanding the Divisibility Rule of 9

To determine if a number is divisible by 9, we use a simple rule: a number is divisible by 9 if the sum of its individual digits is divisible by 9.

Applying the Rule to the 7-Digit Number 48397k5

The given 7-digit number is 48397k5. We need to find the smallest non-zero digit 'k' that makes this number divisible by 9.

• First, let's sum the known digits of the number: $4 + 8 + 3 + 9 + 7 + 5$.
• Calculating the sum: $4 + 8 = 12$, $12 + 3 = 15$, $15 + 9 = 24$, $24 + 7 = 31$, $31 + 5 = 36$.
• So, the sum of the known digits is 36.
• Now, we include the unknown digit 'k'. The total sum of the digits is $36 + k$.
• According to the divisibility rule of 9, this sum ($36 + k$) must be a multiple of 9.

Determining the Value of k

The digit 'k' must be a single digit, meaning its value can range from 0 to 9 ($0 \le k \le 9$).

We need the sum $36 + k$ to be a multiple of 9. Let's look at the multiples of 9:

• $9 \times 1 = 9$
• $9 \times 2 = 18$
• $9 \times 3 = 27$
• $9 \times 4 = 36$
• $9 \times 5 = 45$
• $9 \times 6 = 54$
• ...and so on.

We already have a sum of 36 from the known digits. Since 36 itself is divisible by 9 ($36 = 9 \times 4$), we need to find a value for 'k' such that $36 + k$ is the *next* multiple of 9, or remains a multiple of 9.

Let's test values for k:

• If $k=0$, the sum is $36 + 0 = 36$. 36 is divisible by 9.
• If $k=9$, the sum is $36 + 9 = 45$. 45 is divisible by 9 ($45 = 9 \times 5$).

The question asks for the smallest non-zero value of k.

• The possible values for k that make $36+k$ divisible by 9 are 0 and 9.
• The non-zero values among these are just 9.
• Therefore, the smallest non-zero value for k is 9.

Final Answer Conclusion

For the 7-digit number 48397k5 to be divisible by 9, the sum of its digits must be a multiple of 9. We found that the sum of the known digits is 36. The smallest non-zero digit 'k' that makes the total sum ($36+k$) divisible by 9 is 9.