If `53^(53)-33^(3)` is divided by 10, then the remainder obtained is

Answer

To find the remainder when \( 53^{53} - 33^3 \) is divided by 10, we can use modular arithmetic. Here’s a step-by-step solution:

### Step 1: Calculate \( 53^{53} \mod 10 \)

First, we simplify \( 53 \mod 10 \):
\[ 53 \equiv 3 \mod 10 \]
Thus, we can rewrite \( 53^{53} \) as:
\[ 53^{53} \equiv 3^{53} \mod 10 \]

### Step 2: Find the pattern in \( 3^n \mod 10 \)

Next, we need to find \( 3^{53} \mod 10 \). We can observe the powers of 3 modulo 10:
- \( 3^1 \equiv 3 \mod 10 \)
- \( 3^2 \equiv 9 \mod 10 \)
- \( 3^3 \equiv 27 \equiv 7 \mod 10 \)
- \( 3^4 \equiv 81 \equiv 1 \mod 10 \)

The pattern repeats every 4 terms: \( 3, 9, 7, 1 \).

### Step 3: Determine the exponent modulo 4

To find \( 3^{53} \mod 10 \), we need to determine \( 53 \mod 4 \):
\[ 53 \div 4 = 13 \quad \text{(remainder 1)} \]
Thus, \( 53 \equiv 1 \mod 4 \).

### Step 4: Use the pattern to find \( 3^{53} \mod 10 \)

From the pattern, since \( 53 \equiv 1 \mod 4 \):
\[ 3^{53} \equiv 3^1 \equiv 3 \mod 10 \]

### Step 5: Calculate \( 33^3 \mod 10 \)

Now, we simplify \( 33 \mod 10 \):
\[ 33 \equiv 3 \mod 10 \]
Thus, we have:
\[ 33^3 \equiv 3^3 \mod 10 \]
From our previous calculations:
\[ 3^3 \equiv 7 \mod 10 \]

### Step 6: Combine results

Now we can substitute back into our original expression:
\[ 53^{53} - 33^3 \equiv 3 - 7 \mod 10 \]
Calculating this gives:
\[ 3 - 7 \equiv -4 \mod 10 \]
To convert \(-4\) into a positive remainder, we add \(10\):
\[ -4 + 10 = 6 \]

### Final Answer

Thus, the remainder when \( 53^{53} - 33^3 \) is divided by \( 10 \) is:
\[ \boxed{6} \]