fractional part of `2^(78) /31` is:

Options

A. `(2)/(31)`
B. `(4)/(31)`
C. (8)/(31)`
D. (16)/(31)`

Answer

To find the fractional part of \( \frac{2^{78}}{31} \), we can follow these steps:

### Step 1: Rewrite \( 2^{78} \)
We can express \( 2^{78} \) as \( 2^{3} \cdot 2^{75} \). This helps us to separate the powers of 2.

**Hint:** Factor the exponent to make calculations easier.

### Step 2: Further Break Down \( 2^{75} \)
Next, we can rewrite \( 2^{75} \) as \( (2^5)^{15} \) or \( 32^{15} \). This is useful because \( 32 \) is a power of 2 that we can work with.

**Hint:** Use properties of exponents to simplify the expression.

### Step 3: Use Binomial Expansion
We can express \( 2^{75} \) in a form that allows us to apply the binomial theorem. We can write \( 2^{75} = (8 \cdot 31 + 1)^{15} \).

**Hint:** Recognize that you can use the binomial theorem on expressions of the form \( (a + b)^n \).

### Step 4: Apply the Binomial Theorem
Using the binomial expansion on \( (8 \cdot 31 + 1)^{15} \):
\[ (8 \cdot 31 + 1)^{15} = \sum_{k=0}^{15} \binom{15}{k} (8 \cdot 31)^k (1)^{15-k} \]
This expands to:
\[ \binom{15}{0} (8 \cdot 31)^{0} + \binom{15}{1} (8 \cdot 31)^{1} + \ldots + \binom{15}{15} (1)^{15} \]
The first term is \( 1 \) and all other terms contain \( 31 \) as a factor.

**Hint:** Remember that the binomial expansion gives you a series of terms, where only the last term will not have \( 31 \) as a factor.

### Step 5: Identify the Non-Multiple of 31
From the expansion, we can see that all terms except the last term contribute to multiples of \( 31 \). The last term is \( 1 \).

Thus, we can conclude:
\[ 2^{78} = 8 \cdot (31 \cdots) + 1 \]
This means:
\[ \frac{2^{78}}{31} = 8 \cdots + \frac{1}{31} \]

**Hint:** Identify the integer and fractional parts from the division.

### Step 6: Find the Fractional Part
The integer part of \( \frac{2^{78}}{31} \) is \( 8 \), and the fractional part is \( \frac{1}{31} \).

Thus, the fractional part of \( \frac{2^{78}}{31} \) is:
\[ \frac{1}{31} \]

### Final Answer
The fractional part of \( \frac{2^{78}}{31} \) is \( \frac{1}{31} \).

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