In a four digit number 5a3b `agtb` and `a=b^(3)` .Then the differecne of the number and its cube root is

Options

A. 5850
B. 5220
C. 5256
D. 5814

Explanation

To solve the problem step by step, we need to find the four-digit number \( 5a3b \) given the conditions \( a > b \) and \( a = b^3 \). Then we will calculate the difference between this number and its cube root.

### Step 1: Identify the digits \( a \) and \( b \)
We know that \( a \) and \( b \) are digits, meaning they can take values from 0 to 9. The conditions we have are:
1. \( a > b \)
2. \( a = b^3 \)

### Step 2: Find possible values for \( b \)
Let's check the possible values for \( b \):
- If \( b = 0 \): \( a = 0^3 = 0 \) (not valid since \( a \) must be greater than \( b \))
- If \( b = 1 \): \( a = 1^3 = 1 \) (not valid since \( a \) must be greater than \( b \))
- If \( b = 2 \): \( a = 2^3 = 8 \) (valid since \( 8 > 2 \))
- If \( b = 3 \): \( a = 3^3 = 27 \) (not valid since \( a \) must be a single digit)

The only valid pair is \( a = 8 \) and \( b = 2 \).

### Step 3: Form the four-digit number
Now that we have \( a \) and \( b \):
- The four-digit number \( 5a3b = 5832 \).

### Step 4: Calculate the cube root of the number
Next, we need to find the cube root of \( 5832 \):
1. Factor \( 5832 \):
- \( 5832 \div 2 = 2916 \)
- \( 2916 \div 2 = 1458 \)
- \( 1458 \div 2 = 729 \)
- \( 729 \div 3 = 243 \)
- \( 243 \div 3 = 81 \)
- \( 81 \div 3 = 27 \)
- \( 27 \div 3 = 9 \)
- \( 9 \div 3 = 3 \)
- \( 3 \div 3 = 1 \)

Therefore, the prime factorization of \( 5832 \) is:
\[ 5832 = 2^3 \times 3^6 \]

2. Now, calculate the cube root:
\[ \sqrt[3]{5832} = \sqrt[3]{2^3 \times 3^6} = 2^{3/3} \times 3^{6/3} = 2^1 \times 3^2 = 2 \times 9 = 18 \]

### Step 5: Find the difference between the number and its cube root
Now we find the difference:
\[ 5832 - 18 = 5814 \]

### Final Answer
The difference between the number \( 5832 \) and its cube root \( 18 \) is \( 5814 \).

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