Fractional part of the number 4²⁰²²/15 is equal to

Q: Wait, why are we looking for the remainder? I thought the fractional part was just the decimal part.

That is a great observation! The fractional part of a number x, denoted by {x}, is defined as x - [x], where [x] is the greatest integer less than or equal to x. For a rational number like \frac{N}{D}, we can write it as Q + \frac{R}{D}, where Q is the quotient and R is the remainder. Since Q is an integer, the fractional part is simply \frac{R}{D}. So, finding the remainder is exactly the right path!

Q: Why did we choose 4^2 and not 4^3?

We choose 4^2 because 16 is very close to our divisor, 15. The Binomial Theorem works best when we can express our base as (divisor + 1) or (divisor - 1). Since 16 = 15 + 1, it allows us to use the expansion (15 + 1)^n, where every term except the last one will be a multiple of 15.

Q: Is the remainder always 1 in these types of problems?

Not always! It depends on the base and the divisor. In this specific case, because we could write the base as (15 + 1), the remainder turned out to be 1. If the base were different, say 3, we would look for a power of 3 close to a multiple of 15. Always look for that 'plus or minus one' relationship to simplify the problem.

Question

Fractional part of the number $\frac{4^{2022}}{15}$ is equal to

Options:

(A) $\frac{8}{15}$

(B) $\frac{4}{15}$

(C) $\frac{14}{15}$

(D) $\frac{1}{15}$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are staring at the expression $\frac{4^{2022}}{15}$. At first glance, it looks like a nightmare.

Four raised to the power of two thousand twenty-two is a number so gargantuan that it would fill entire books if written out in full. How could anyone possibly find its fractional part?

This is where the beauty of mathematics shines. We do not need to calculate the number; we only need to understand its structure.

### Demystifying the Fractional Part

Let us start with the definition. The fractional part of a number $x$, denoted by $\{x\}$, is simply $x - [x]$, where $[x]$ is the greatest integer less than or equal to $x$.

When we have a fraction $\frac{N}{D}$, we can perform long division to write it as $Q + \frac{R}{D}$, where $Q$ is the integer quotient and $R$ is the remainder. Since $Q$ is an integer, the fractional part is simply $\frac{R}{D}$.

Our mission, therefore, is not to calculate the massive numerator, but to find the remainder when $4^{2022}$ is divided by $15$.

### The Power of Proximity

We need a strategy. We are looking for a remainder, and we have a base of $4$ and a divisor of $15$.

Is there a power of $4$ that is close to $15$? Let us test: $4^1 = 4$, and $4^2 = 16$.

Eureka! $16$ is just $15 + 1$. This is the key that unlocks the entire problem. By rewriting our base, we can transform this impossible calculation into a simple algebraic expansion.

### The Binomial Magic

We rewrite $4^{2022}$ as $(4^2)^{1011}$, which is $16^{1011}$. Now, we substitute $16$ with $(15 + 1)$.

Our expression becomes $(15 + 1)^{1011}$. Now, we invoke the Binomial Theorem:

$$ (a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r $$

When we expand $(15 + 1)^{1011}$, every single term contains a factor of $15$, except for the very last term, which is $\binom{1011}{1011} 1^{1011}$.

We can group all the terms containing $15$ and call their sum $15k$, where $k$ is some integer. Thus:

$$ 4^{2022} = 15k + 1 $$

### The Final Victory

Now, we return to our original fraction:

$$ \frac{4^{2022}}{15} = \frac{15k + 1}{15} $$

Splitting this, we get:

$$ \frac{15k}{15} + \frac{1}{15} = k + \frac{1}{15} $$

Since $k$ is an integer, the fractional part is simply $\frac{1}{15}$.

Look at that! We navigated through a sea of exponents and binomial coefficients to find a result as simple as **$\frac{1}{15}$**. This is the power of mathematical thinking: turning the impossible into the elegant.

Animated Solution for Mathematics - Binomial Theorem: Fractional part of the number $\frac{4 ^{2022}}{15}$ is equal to

Select Answer:

Visualized Solution

Understanding Fractional Part

Fractional Part of a Rational Number

Setting Up the Remainder Problem

Strategic Base Transformation

Applying Exponent Rules

Preparing for Binomial Expansion

The Binomial Expansion Tool

Identifying the Remainder

Simplifying the Last Term

Final Calculation

The Sigma Insight: Binomial Expansion for Positive Integral Index

Analyzing the Setup

Demystifying the Fractional Part

The Power of Proximity

The Binomial Magic

The Final Victory

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Animated Solution for Mathematics - Binomial Theorem: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Fractional part of the number 1542022​ is equal to