Ram's age was square of a number last year and it will be cube of a number next year. How long must he wait before his age is again the cube of a number?

Options

A. 39 years
B. 10 years
C. 38 years
D. 64 years

Explanation

To solve the problem step by step, let's break it down:

### Step 1: Define Ram's current age
Let Ram's current age be \( x \).

### Step 2: Establish the conditions from the problem
According to the problem:
- Last year, Ram's age was \( x - 1 \), which is a perfect square.
- Next year, Ram's age will be \( x + 1 \), which is a perfect cube.

### Step 3: Set up the equations
From the above conditions, we can say:
- \( x - 1 = n^2 \) (where \( n \) is some integer)
- \( x + 1 = m^3 \) (where \( m \) is some integer)

### Step 4: Relate the two equations
From the two equations, we can express \( x \) in terms of \( n \) and \( m \):
1. \( x = n^2 + 1 \)
2. \( x = m^3 - 1 \)

### Step 5: Set the equations equal to each other
Since both expressions equal \( x \), we can set them equal:
\[ n^2 + 1 = m^3 - 1 \]
This simplifies to:
\[ m^3 - n^2 = 2 \]

### Step 6: Find suitable values for \( n \) and \( m \)
We need to find integers \( n \) and \( m \) such that the difference between the cube of \( m \) and the square of \( n \) is 2. We can test small values for \( m \):
- If \( m = 3 \):
\[ 3^3 = 27 \]
\[ n^2 = 27 - 2 = 25 \Rightarrow n = 5 \]

So, we have:
- \( n = 5 \) (which gives \( n^2 = 25 \))
- \( m = 3 \) (which gives \( m^3 = 27 \))

### Step 7: Calculate Ram's current age
Using \( n \):
\[ x = n^2 + 1 = 25 + 1 = 26 \]

### Step 8: Determine when Ram's age will be a cube again
The next cube after \( 27 \) (which is \( 3^3 \)) is \( 64 \) (which is \( 4^3 \)).

### Step 9: Calculate how long Ram must wait
To find out how long Ram must wait to reach \( 64 \):
\[ 64 - 26 = 38 \]

### Conclusion
Ram must wait **38 years** before his age is again the cube of a number.