What will be the remainder when `19^(100)` is divided by 20?

Options

A. 19
B. 20
C. 3
D. 1

Answer

To find the remainder when \( 19^{100} \) is divided by 20, we can follow these steps:

### Step 1: Rewrite the base
We can express \( 19 \) in a form that is easier to work with in relation to \( 20 \):
\[ 19 = 20 - 1 \]
Thus, we can rewrite \( 19^{100} \) as:
\[ 19^{100} = (20 - 1)^{100} \]

### Step 2: Apply the Binomial Theorem
Using the Binomial Theorem, we can expand \( (20 - 1)^{100} \):
\[ (20 - 1)^{100} = \sum_{k=0}^{100} \binom{100}{k} \cdot 20^k \cdot (-1)^{100-k} \]
However, since we are interested in the remainder when divided by \( 20 \), we only need to consider the term where \( k = 0 \) (the constant term), because all other terms will be multiples of \( 20 \) and will not affect the remainder.

### Step 3: Identify the constant term
The constant term occurs when \( k = 0 \):
\[ \binom{100}{0} \cdot 20^0 \cdot (-1)^{100} = 1 \cdot 1 \cdot 1 = 1 \]

### Step 4: Calculate the remainder
Now, we have:
\[ (20 - 1)^{100} \equiv 1 \mod 20 \]
Thus, the remainder when \( 19^{100} \) is divided by \( 20 \) is:
\[ \text{Remainder} = 1 \]

### Final Answer
Therefore, the remainder when \( 19^{100} \) is divided by \( 20 \) is \( 1 \).

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