Fill in the blanck and select the correct option.
A number ending in `ul((P))` number of zeroes is never a perfect square.
(ii) the square of `ul((Q))` natural number can always be written as the sum of two consecutive positive integers.
(iii) The sum of the first n odd natural numbers is `ul((R))` .
(iv) if `(3xx3xx7)^(@)=3969` then `sqrt(3969)=ul((S))`

Options

A. `{:(P" "Q" "R" "S),("Odd"" ""even"" "2n" "62):}`
B. `{:(P" "Q" "R" "S),("Even"" ""odd"" "n^(2)" "69):}`
C. `{:(P" "Q" "R" "S),("Even"" ""even"" "n^(3)" "39):}`
D. `{:(P" "Q" "R" "S),("Odd"" ""odd"" "n^(2)" "63):}`

Answer

Let's solve the question step by step.

### Step 1: Determine the value of P
The statement is: "A number ending in `ul((P))` number of zeroes is never a perfect square."

To determine the value of P, we need to understand the condition for a number to be a perfect square. A number can only be a perfect square if it has an even number of prime factors. If a number ends in an odd number of zeroes, it means it has an odd number of factors of 10 (which is 2 and 5). Therefore, a number ending in an odd number of zeroes cannot be a perfect square.

**Value of P:** Odd

**Hint for Step 1:** Think about the prime factorization of numbers and how the number of zeroes relates to the factors of 10.

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### Step 2: Determine the value of Q
The statement is: "The square of `ul((Q))` natural number can always be written as the sum of two consecutive positive integers."

To find Q, we can analyze the squares of natural numbers. The square of any odd natural number can be expressed as the sum of two consecutive integers. For example:
- \(3^2 = 9 = 4 + 5\)
- \(5^2 = 25 = 12 + 13\)

Thus, we can conclude that Q must be an odd natural number.

**Value of Q:** Odd

**Hint for Step 2:** Consider how squares of odd numbers can be represented as sums of two consecutive integers.

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### Step 3: Determine the value of R
The statement is: "The sum of the first n odd natural numbers is `ul((R))`."

The sum of the first n odd natural numbers is known to be \(n^2\). This can be derived from the formula for the sum of an arithmetic series or can be observed through examples:
- Sum of first 1 odd number: \(1 = 1^2\)
- Sum of first 2 odd numbers: \(1 + 3 = 4 = 2^2\)
- Sum of first 3 odd numbers: \(1 + 3 + 5 = 9 = 3^2\)

Thus, the sum of the first n odd natural numbers is \(n^2\).

**Value of R:** \(n^2\)

**Hint for Step 3:** Recall the formula for the sum of an arithmetic series and how it applies to odd numbers.

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### Step 4: Determine the value of S
The statement is: "If `(3xx3xx7)^(@)=3969` then `sqrt(3969)=ul((S))`."

To find S, we need to calculate the square root of 3969.

First, we can factor 3969:
- \(3969 = 63 \times 63\) (since \(63^2 = 3969\))

Thus, the square root of 3969 is 63.

**Value of S:** 63

**Hint for Step 4:** Use prime factorization or direct calculation to find the square root of the number.

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### Final Answers:
- P: Odd
- Q: Odd
- R: \(n^2\)
- S: 63