Two numbers are randomly selected from the first 100 natural numbers. The probability the product is divisible by 17 is

Options

A. `931/990`
B. `97/990`
C. `101/990`
D. `879/990`

Answer

To find the probability that the product of two randomly selected numbers from the first 100 natural numbers is divisible by 17, we can follow these steps:

### Step 1: Identify the total number of natural numbers
The first 100 natural numbers are: 1, 2, 3, ..., 100. Therefore, the total number of natural numbers is 100.

### Step 2: Identify the numbers divisible by 17
Next, we need to find how many numbers from 1 to 100 are divisible by 17. The multiples of 17 within this range are:
- 17 (1 × 17)
- 34 (2 × 17)
- 51 (3 × 17)
- 68 (4 × 17)
- 85 (5 × 17)

Thus, there are 5 numbers that are divisible by 17.

### Step 3: Calculate the numbers not divisible by 17
To find the numbers that are not divisible by 17, we subtract the count of numbers divisible by 17 from the total count of natural numbers:
\[ 100 - 5 = 95 \]
So, there are 95 numbers that are not divisible by 17.

### Step 4: Calculate the total ways to choose 2 numbers
The total ways to choose 2 numbers from 100 is given by the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Thus, the total ways to choose 2 numbers from 100 is:
\[ \binom{100}{2} = \frac{100 \times 99}{2} = 4950 \]

### Step 5: Calculate the ways to choose 2 numbers not divisible by 17
The total ways to choose 2 numbers from the 95 numbers that are not divisible by 17 is:
\[ \binom{95}{2} = \frac{95 \times 94}{2} = 4465 \]

### Step 6: Calculate the probability that the product is not divisible by 17
The probability that the product of the two selected numbers is not divisible by 17 is given by the ratio of the favorable outcomes to the total outcomes:
\[ P(\text{not divisible by 17}) = \frac{\text{Ways to choose 2 not divisible by 17}}{\text{Total ways to choose 2}} = \frac{4465}{4950} \]

### Step 7: Calculate the probability that the product is divisible by 17
To find the probability that the product is divisible by 17, we subtract the probability that the product is not divisible by 17 from 1:
\[ P(\text{divisible by 17}) = 1 - P(\text{not divisible by 17}) = 1 - \frac{4465}{4950} \]
Calculating this gives:
\[ P(\text{divisible by 17}) = \frac{4950 - 4465}{4950} = \frac{485}{4950} \]

### Step 8: Simplify the fraction
Now we can simplify the fraction:
\[ P(\text{divisible by 17}) = \frac{485}{4950} = \frac{97}{990} \]

Thus, the final answer is:
\[ \text{The probability that the product is divisible by 17 is } \frac{97}{990}. \]