Finding Remainder of $98!$ divided by $101$
We need to find the remainder when $98!$ is divided by $101$. We can use Wilson's Theorem for this problem.
Wilson's Theorem Application
Wilson's Theorem states that for any prime number $p$, we have:
$ (p-1)! \equiv -1 \pmod{p} $
In this case, $p = 101$, which is a prime number. Applying Wilson's Theorem:
$ 100! \equiv -1 \pmod{101} $
Relating $100!$ to $98!$
We can write $100!$ as:
$ 100! = 100 \times 99 \times 98! $
Substitute this back into the congruence from Wilson's Theorem:
$ (100 \times 99 \times 98!) \equiv -1 \pmod{101} $
Simplifying the Congruence
We know that $100 \equiv -1 \pmod{101}$ and $99 \equiv -2 \pmod{101}$. Substituting these values:
$ (-1 \times -2 \times 98!) \equiv -1 \pmod{101} $
$ (2 \times 98!) \equiv -1 \pmod{101} $
Since $-1 \equiv 100 \pmod{101}$, we have:
$ 2 \times 98! \equiv 100 \pmod{101} $
Solving for $98!$
To find the value of $98!$, we need to solve the congruence $2 \times 98! \equiv 100 \pmod{101}$. This involves finding the modular multiplicative inverse of $2$ modulo $101$.
We need to find a number $x$ such that $2x \equiv 1 \pmod{101}$. We can see that $2 \times 51 = 102$, and $102 \equiv 1 \pmod{101}$. So, the modular inverse of $2$ modulo $101$ is $51$.
Multiply both sides of the congruence $2 \times 98! \equiv 100 \pmod{101}$ by $51$:
$ 51 \times (2 \times 98!) \equiv 51 \times 100 \pmod{101} $
$ (51 \times 2) \times 98! \equiv 5100 \pmod{101} $
$ 1 \times 98! \equiv 5100 \pmod{101} $
Now, we find the remainder of $5100$ when divided by $101$:
$ 5100 = 50 \times 101 + 50 $
Therefore, $5100 \equiv 50 \pmod{101}$.
So, the final result is:
$ 98! \equiv 50 \pmod{101} $
The remainder when $98!$ is divided by $101$ is $50$. This value falls within the given range of $49.9$ and $50.1$.
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 __________ - Let $p_1$ and $p_2$ denote two arbitrary prime numbers. Which one of the following statements is correct for all values of $p_1$ and $p_2$?
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:- 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 ________.
