The least number, which 715 must be multiplied with in order to get a multiple of 825 is-

Options

A. 10
B. 12
C. 15
D. 35

Answer

To find the least number which 715 must be multiplied with in order to get a multiple of 825, we can follow these steps:

### Step 1: Set up the equation
Let the least number be \( X \). We want to find \( X \) such that:
\[ 715 \times X = n \times 825 \]
where \( n \) is some integer.

### Step 2: Rearrange the equation
From the equation above, we can express \( X \) as:
\[ X = \frac{n \times 825}{715} \]

### Step 3: Simplify the fraction
To find the least value of \( X \), we need to simplify \( \frac{825}{715} \). We can do this by finding the greatest common divisor (GCD) of 825 and 715.

### Step 4: Find the GCD of 825 and 715
To find the GCD, we can use the prime factorization method or the Euclidean algorithm.

- **Prime Factorization**:
- \( 825 = 3 \times 5^2 \times 11 \)
- \( 715 = 5 \times 11 \times 13 \)

The common factors are \( 5 \) and \( 11 \). Therefore, the GCD is:
\[ GCD(825, 715) = 5 \times 11 = 55 \]

### Step 5: Divide both numbers by the GCD
Now we can simplify \( \frac{825}{715} \):
\[ \frac{825 \div 55}{715 \div 55} = \frac{15}{13} \]

### Step 6: Express \( X \)
Substituting back, we have:
\[ X = \frac{n \times 825}{715} = n \times \frac{15}{13} \]

### Step 7: Find the least integer value of \( n \)
To make \( X \) an integer, \( n \) must be a multiple of \( 13 \). The smallest value for \( n \) is \( 13 \).

### Step 8: Calculate \( X \)
Substituting \( n = 13 \):
\[ X = 13 \times \frac{15}{13} = 15 \]

### Conclusion
Thus, the least number which 715 must be multiplied with in order to get a multiple of 825 is:
\[ \boxed{15} \]

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