A three digit number is such that this number itself is divisible by the sum of its digits. The sum of hundreds and unit digit is 6 while the sum of the tens and unit digit is 5. What is the ratio of unit and tens digit:

Options

A. a. `5:3
B. b. `2:3`
C. c. `3:4`
D. d. `2:7`

Answer

To solve the problem step by step, we need to find a three-digit number represented as XYZ, where:

- X is the hundreds digit,
- Y is the tens digit,
- Z is the units digit.

We are given two conditions:
1. The sum of the hundreds and units digits (X + Z) is 6.
2. The sum of the tens and units digits (Y + Z) is 5.

### Step 1: Set Up the Equations
From the given conditions, we can write two equations:
1. \( X + Z = 6 \) (Equation 1)
2. \( Y + Z = 5 \) (Equation 2)

### Step 2: Express Y in Terms of Z
From Equation 2, we can express Y in terms of Z:
\[ Y = 5 - Z \] (Equation 3)

### Step 3: Substitute Y in Terms of Z into Equation 1
Now, we can substitute Equation 3 into Equation 1 to find possible values for X and Z:
\[ X + Z = 6 \]
We can rearrange this to find X:
\[ X = 6 - Z \] (Equation 4)

### Step 4: Find Possible Values for X, Y, and Z
Now we have:
- \( X = 6 - Z \)
- \( Y = 5 - Z \)

Since X, Y, and Z are digits (0-9), we need to find values of Z that keep X and Y as valid digits.

### Step 5: Determine Valid Values for Z
Let’s analyze the possible values for Z:
- If \( Z = 0 \):
- \( X = 6 - 0 = 6 \)
- \( Y = 5 - 0 = 5 \)
- Number: 650 (valid)

- If \( Z = 1 \):
- \( X = 6 - 1 = 5 \)
- \( Y = 5 - 1 = 4 \)
- Number: 541 (valid)

- If \( Z = 2 \):
- \( X = 6 - 2 = 4 \)
- \( Y = 5 - 2 = 3 \)
- Number: 432 (valid)

- If \( Z = 3 \):
- \( X = 6 - 3 = 3 \)
- \( Y = 5 - 3 = 2 \)
- Number: 321 (valid)

- If \( Z = 4 \):
- \( X = 6 - 4 = 2 \)
- \( Y = 5 - 4 = 1 \)
- Number: 214 (valid)

- If \( Z = 5 \):
- \( X = 6 - 5 = 1 \)
- \( Y = 5 - 5 = 0 \)
- Number: 105 (valid)

- If \( Z \geq 6 \):
- X would become negative, which is invalid.

### Step 6: Check Divisibility Condition
Now we need to check if these numbers are divisible by the sum of their digits:
- For 650: \( 6 + 5 + 0 = 11 \) (not divisible)
- For 541: \( 5 + 4 + 1 = 10 \) (not divisible)
- For 432: \( 4 + 3 + 2 = 9 \) (divisible)
- For 321: \( 3 + 2 + 1 = 6 \) (divisible)
- For 214: \( 2 + 1 + 4 = 7 \) (not divisible)
- For 105: \( 1 + 0 + 5 = 6 \) (divisible)

### Step 7: Valid Numbers
The valid numbers that meet all conditions are:
- 432
- 321
- 105

### Step 8: Find the Ratio of Unit and Tens Digits
Now we need to find the ratio of the unit digit (Z) to the tens digit (Y) for these valid numbers:
- For 432: \( Z = 2, Y = 3 \) → Ratio = \( \frac{2}{3} \)
- For 321: \( Z = 1, Y = 2 \) → Ratio = \( \frac{1}{2} \)
- For 105: \( Z = 5, Y = 0 \) → Ratio is undefined (since Y cannot be 0)

### Conclusion
The valid ratios of unit digit to tens digit for the valid numbers are \( \frac{2}{3} \) and \( \frac{1}{2} \).

### Final Answer
Since the question asks for the ratio of the unit and tens digit, we can conclude that the valid ratios are \( \frac{2}{3} \) or \( \frac{1}{2} \).