Find the remainder when `(2)^(51)` is divided by 5.

Options

A. 8
B. 7
C. 4
D. 5

Answer

To find the remainder when \(2^{51}\) is divided by 5, we can use modular arithmetic. Here’s a step-by-step solution:

### Step 1: Identify the expression
We need to find \(2^{51} \mod 5\).

### Step 2: Use Fermat's Little Theorem
According to Fermat's Little Theorem, if \(p\) is a prime number and \(a\) is an integer not divisible by \(p\), then:
\[ a^{p-1} \equiv 1 \mod p \]
In our case, \(a = 2\) and \(p = 5\). Since 2 is not divisible by 5, we can apply the theorem:
\[ 2^{5-1} = 2^4 \equiv 1 \mod 5 \]

### Step 3: Reduce the exponent
Now, we can reduce the exponent \(51\) modulo \(4\) (since \(2^4 \equiv 1\)):
\[ 51 \mod 4 \]
Calculating \(51 \div 4\) gives a quotient of \(12\) and a remainder of \(3\). Therefore:
\[ 51 \equiv 3 \mod 4 \]

### Step 4: Substitute back into the expression
Now we can substitute back into our expression:
\[ 2^{51} \equiv 2^3 \mod 5 \]

### Step 5: Calculate \(2^3\)
Now we calculate \(2^3\):
\[ 2^3 = 8 \]

### Step 6: Find \(8 \mod 5\)
Now we find the remainder when \(8\) is divided by \(5\):
\[ 8 \mod 5 = 3 \]

### Final Result
Thus, the remainder when \(2^{51}\) is divided by \(5\) is:
\[ \boxed{3} \]

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