`53^3-46^3-7^3` is divided by:

Options

A. 6 and 9
B. 2 and 21
C. 21 and 23
D. both (b) and(c)

Answer

To solve the expression \( 53^3 - 46^3 - 7^3 \) and determine what it is divisible by, we can use the identity for the difference of cubes.

### Step-by-step Solution:

1. **Identify the terms**: We have the expression \( 53^3 - 46^3 - 7^3 \). Let's denote:
- \( a = 53 \)
- \( b = 46 \)
- \( c = 7 \)

2. **Check the condition for the identity**: We will check if \( a + b + c = 0 \):
\[ 53 + 46 + 7 = 106 \quad (\text{not } 0) \]
Since \( a + b + c \neq 0 \), we cannot directly apply the identity \( a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \).

3. **Rearranging the expression**: We can rearrange the expression as follows:
\[ 53^3 - 46^3 - 7^3 = 53^3 - (46^3 + 7^3) \]

4. **Using the sum of cubes**: We can apply the sum of cubes formula on \( 46^3 + 7^3 \):
\[ x^3 + y^3 = (x + y)(x^2 - xy + y^2) \]
where \( x = 46 \) and \( y = 7 \):
\[ 46^3 + 7^3 = (46 + 7)(46^2 - 46 \cdot 7 + 7^2) = 53(46^2 - 322 + 49) \]

5. **Calculating \( 46^2 - 46 \cdot 7 + 7^2 \)**:
- \( 46^2 = 2116 \)
- \( 46 \cdot 7 = 322 \)
- \( 7^2 = 49 \)
- Therefore:
\[ 46^2 - 46 \cdot 7 + 7^2 = 2116 - 322 + 49 = 1843 \]
Thus, \( 46^3 + 7^3 = 53 \cdot 1843 \).

6. **Substituting back**: Now we substitute back into our expression:
\[ 53^3 - (46^3 + 7^3) = 53^3 - 53 \cdot 1843 \]

7. **Factoring out \( 53 \)**:
\[ = 53(53^2 - 1843) \]

8. **Calculating \( 53^2 \)**:
\[ 53^2 = 2809 \]
Therefore:
\[ 53^2 - 1843 = 2809 - 1843 = 966 \]

9. **Final expression**: Thus, we have:
\[ 53^3 - 46^3 - 7^3 = 53 \cdot 966 \]

10. **Finding the divisors**: Since the expression is \( 53 \cdot 966 \), it is divisible by both \( 53 \) and \( 966 \).

### Conclusion:
The expression \( 53^3 - 46^3 - 7^3 \) is divisible by \( 53 \) and \( 966 \).