Two numbers a and b are chosen at random from the set of first 30 natural numbers. The probability that `a^(2)-b^(2)` is divisible by 3 is:

Options

A. `37/87`
B. `47/87`
C. `17/29`
D. None of these

Answer

To solve the problem of finding the probability that \( a^2 - b^2 \) is divisible by 3 when two numbers \( a \) and \( b \) are chosen from the first 30 natural numbers, we can follow these steps:

### Step 1: Understanding the Expression
The expression \( a^2 - b^2 \) can be factored as \( (a - b)(a + b) \). For this product to be divisible by 3, at least one of the factors \( a - b \) or \( a + b \) must be divisible by 3.

### Step 2: Classifying Numbers Modulo 3
The first 30 natural numbers can be classified based on their remainders when divided by 3:
- Numbers congruent to 0 mod 3: \( 3, 6, 9, \ldots, 30 \) (10 numbers)
- Numbers congruent to 1 mod 3: \( 1, 4, 7, \ldots, 28 \) (10 numbers)
- Numbers congruent to 2 mod 3: \( 2, 5, 8, \ldots, 29 \) (10 numbers)

### Step 3: Total Ways to Choose Two Numbers
The total number of ways to choose 2 numbers from 30 is given by the combination formula:
\[ \binom{30}{2} = \frac{30 \times 29}{2} = 435 \]

### Step 4: Finding Favorable Outcomes
We need to count the favorable outcomes where \( a^2 - b^2 \) is divisible by 3. This occurs in the following cases:
1. Both \( a \) and \( b \) are congruent to 0 mod 3.
2. Both \( a \) and \( b \) are congruent to 1 mod 3.
3. Both \( a \) and \( b \) are congruent to 2 mod 3.
4. One is congruent to 1 mod 3 and the other is congruent to 2 mod 3.

Calculating these:
- Case 1: Choosing 2 from 10 numbers (0 mod 3):
\[ \binom{10}{2} = \frac{10 \times 9}{2} = 45 \]
- Case 2: Choosing 2 from 10 numbers (1 mod 3):
\[ \binom{10}{2} = 45 \]
- Case 3: Choosing 2 from 10 numbers (2 mod 3):
\[ \binom{10}{2} = 45 \]
- Case 4: Choosing 1 from 10 numbers (1 mod 3) and 1 from 10 numbers (2 mod 3):
\[ 10 \times 10 = 100 \]

### Step 5: Total Favorable Outcomes
Now, we sum the favorable outcomes:
\[ 45 + 45 + 45 + 100 = 235 \]

### Step 6: Calculating the Probability
The probability \( P \) that \( a^2 - b^2 \) is divisible by 3 is given by the ratio of favorable outcomes to total outcomes:
\[ P = \frac{235}{435} \]

### Step 7: Simplifying the Probability
We can simplify this fraction by dividing both the numerator and the denominator by 5:
\[ P = \frac{47}{87} \]

### Final Answer
The probability that \( a^2 - b^2 \) is divisible by 3 is:
\[ \frac{47}{87} \]