What is the remainder when we divide `4^(50) + 7^(50)` by 65 ?

Options

A. 1
B. 2
C. 0
D. 3

Answer

To find the remainder when \( 4^{50} + 7^{50} \) is divided by 65, we can use properties of modular arithmetic. Here’s a step-by-step solution:

### Step 1: Rewrite the expression
We start with the expression:
\[ 4^{50} + 7^{50} \]

### Step 2: Use properties of exponents
Notice that both 4 and 7 can be expressed in terms of their squares:
\[ 4^{50} = (4^2)^{25} = 16^{25} \]
\[ 7^{50} = (7^2)^{25} = 49^{25} \]
So we can rewrite the expression as:
\[ 16^{25} + 49^{25} \]

### Step 3: Use the sum of powers theorem
According to the theorem, if \( n \) is odd, then:
\[ a^n + b^n \equiv 0 \mod (a + b) \]
In our case, \( n = 25 \) (which is odd), \( a = 16 \), and \( b = 49 \). Therefore:
\[ 16^{25} + 49^{25} \equiv 0 \mod (16 + 49) \]
Calculating \( 16 + 49 \):
\[ 16 + 49 = 65 \]
This means:
\[ 16^{25} + 49^{25} \equiv 0 \mod 65 \]

### Step 4: Conclusion
Since \( 4^{50} + 7^{50} \equiv 0 \mod 65 \), the remainder when \( 4^{50} + 7^{50} \) is divided by 65 is:
\[ \boxed{0} \]