Prime Number Properties Analysis
This solution examines the properties of statements involving two arbitrary prime numbers, $p_1$ and $p_2$. A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself.
Evaluating Prime Number Statements
- Statement 1: $p_1 + p_2$ is not a prime number.
- Test Case: Let $p_1 = 2$ and $p_2 = 3$. Both are prime.
- Sum: $p_1 + p_2 = 2 + 3 = 5$.
- Result: 5 is a prime number. Therefore, this statement is false as it's not true for all primes.
- Statement 2: $p_1p_2$ is not a prime number.
- Definition: A prime number has only 1 and itself as divisors.
- Consider the product $p_1p_2$. Since $p_1$ and $p_2$ are primes, they are both greater than 1.
- The divisors of $p_1p_2$ include $1$, $p_1$, $p_2$, and $p_1p_2$.
- As $p_1 > 1$ and $p_2 > 1$, $p_1$ is a divisor other than 1 and $p_1p_2$. Similarly for $p_2$.
- Therefore, $p_1p_2$ always has more than two divisors, making it a composite number.
- Result: This statement is always true.
- Statement 3: $p_1 + p_2 + 1$ is a prime number.
- Test Case: Let $p_1 = 2$ and $p_2 = 3$.
- Sum + 1: $p_1 + p_2 + 1 = 2 + 3 + 1 = 6$.
- Result: 6 is not a prime number. Therefore, this statement is false.
- Statement 4: $p_1p_2 + 1$ is a prime number.
- Test Case: Let $p_1 = 3$ and $p_2 = 5$. Both are prime.
- Product + 1: $p_1p_2 + 1 = (3)(5) + 1 = 15 + 1 = 16$.
- Result: 16 is not a prime number. Therefore, this statement is false.
Correct Statement Identification
The analysis confirms that only the second statement, "$p_1p_2$ is not a prime number", is universally true for any two prime numbers $p_1$ and $p_2$. This is because the product of two primes will always have $p_1$ and $p_2$ as factors, in addition to 1 and the product itself, disqualifying it from being prime.
Important Questions from Numerical Reasoning
- $P, Q, R, S, X$, and $Y$ are distinct single-digit whole numbers taking values from 0 to 9.
$PQ$ is a two-digit number with $Q$ being in the units place and $P$ in the tens place. Similarly, $RS$ is a two-digit number.
It is known that $PQ$ and $RS$ are consecutive numbers and
$(PQ)^2 + (RS)^2 = XYP$, with $XYP$ being a three-digit number.
The value of $Y$ is __________ If $\oplus \div \odot = 2$, $\oplus \div \triangle = 3$, $\odot + \triangle = 5$, and $\Delta \times \otimes = 10$,
then the value of $(\otimes - \oplus)^2$ is:- The remainder when $98!$ is divided by $101$ is equal to ________
- A 'frabjous' number is defined as a 3 digit number with all digits odd, and no two adjacent digits being the same. For example, 137 is a frabjous number, while 133 is not. How many such frabjous numbers exist?
Ankita has to climb 5 stairs starting at the ground, while respecting the following rules:
1. At any stage, Ankita can move either one or two stairs up.
2. At any stage, Ankita cannot move to a lower step.
Let $F(N)$ denote the number of possible ways in which Ankita can reach the $N^{th}$ stair. For example, $F(1) = 1$, $F(2) = 2$, $F(3) = 3$. The value of $F(5)$ is ________.
