The sum of the digits of a 2-digit number is 10. A number is selected at random. Find the probability of the chosen number to be divisible by 3.

Answer

To solve the problem step by step, we will follow the outlined process to find the probability of a randomly selected two-digit number, whose digits sum to 10, being divisible by 3.

### Step 1: Identify the two-digit numbers whose digits sum to 10.

A two-digit number can be represented as \(10a + b\), where \(a\) is the tens digit and \(b\) is the units digit. The condition given is:

\[ a + b = 10 \]

Since \(a\) must be a digit from 1 to 9 (as it is the tens place in a two-digit number), we can find the possible pairs \((a, b)\):

- If \(a = 1\), then \(b = 9\) → Number: 19
- If \(a = 2\), then \(b = 8\) → Number: 28
- If \(a = 3\), then \(b = 7\) → Number: 37
- If \(a = 4\), then \(b = 6\) → Number: 46
- If \(a = 5\), then \(b = 5\) → Number: 55
- If \(a = 6\), then \(b = 4\) → Number: 64
- If \(a = 7\), then \(b = 3\) → Number: 73
- If \(a = 8\), then \(b = 2\) → Number: 82
- If \(a = 9\), then \(b = 1\) → Number: 91

Thus, the two-digit numbers whose digits sum to 10 are: **19, 28, 37, 46, 55, 64, 73, 82, 91**.

### Step 2: Count the total number of outcomes.

The total number of two-digit numbers whose digits sum to 10 is **9**.

### Step 3: Identify which of these numbers are divisible by 3.

To find out which of these numbers are divisible by 3, we will check each number:

- **19**: \(19 \div 3 = 6.33\) (not divisible)
- **28**: \(28 \div 3 = 9.33\) (not divisible)
- **37**: \(37 \div 3 = 12.33\) (not divisible)
- **46**: \(46 \div 3 = 15.33\) (not divisible)
- **55**: \(55 \div 3 = 18.33\) (not divisible)
- **64**: \(64 \div 3 = 21.33\) (not divisible)
- **73**: \(73 \div 3 = 24.33\) (not divisible)
- **82**: \(82 \div 3 = 27.33\) (not divisible)
- **91**: \(91 \div 3 = 30.33\) (not divisible)

None of the numbers are divisible by 3.

### Step 4: Count the favorable outcomes.

The number of favorable outcomes (numbers that are divisible by 3) is **0**.

### Step 5: Calculate the probability.

The probability \(P\) of selecting a number that is divisible by 3 is given by:

\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{0}{9} = 0 \]

### Final Answer:

The probability that the chosen number is divisible by 3 is **0**.

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