When `5^2450` is divided by 126, the remainder obtained is ?

Options

A. 5
B. 25
C. 125
D. 1

Answer

To find the remainder when \( 5^{2450} \) is divided by 126, we can follow these steps:

### Step 1: Simplify the exponent
We can express \( 5^{2450} \) in a more manageable form. Notice that \( 5^3 = 125 \). We can rewrite \( 5^{2450} \) as:
\[ 5^{2450} = (5^3)^{816} \times 5^2 \]
This is because \( 2450 = 3 \times 816 + 2 \).

### Step 2: Calculate \( 5^3 \) modulo 126
Next, we calculate \( 5^3 \) modulo 126:
\[ 5^3 = 125 \]
Now, we find the remainder when 125 is divided by 126:
\[ 125 \mod 126 = 125 \]
This means that \( 125 \equiv -1 \mod 126 \) (since \( 125 + 1 = 126 \)).

### Step 3: Substitute back into the expression
Now we can substitute back into our expression:
\[ 5^{2450} \equiv (-1)^{816} \times 5^2 \mod 126 \]

### Step 4: Evaluate \( (-1)^{816} \)
Since 816 is an even number, we have:
\[ (-1)^{816} = 1 \]
Thus, we can simplify our expression to:
\[ 5^{2450} \equiv 1 \times 5^2 \mod 126 \]

### Step 5: Calculate \( 5^2 \) modulo 126
Now we calculate \( 5^2 \):
\[ 5^2 = 25 \]
So we have:
\[ 5^{2450} \equiv 25 \mod 126 \]

### Step 6: Conclusion
The remainder when \( 5^{2450} \) is divided by 126 is:
\[ \boxed{25} \]