What is the product of the digits of the greatest number that divides 690 and 875 leaving remainders 10 and 25 respectively ?

Options

A. 0
B. 10
C. 14
D. 20

Answer

To solve the problem, we need to find the greatest number that divides 690 and 875, leaving remainders of 10 and 25 respectively. Here’s a step-by-step solution:

### Step 1: Adjust the numbers by subtracting the remainders
To find the number that divides 690 and 875 with the specified remainders, we first subtract the remainders from each number.

- For 690, subtract 10:
\[ 690 - 10 = 680 \]

- For 875, subtract 25:
\[ 875 - 25 = 850 \]

### Step 2: Find the HCF of the adjusted numbers
Now we need to find the highest common factor (HCF) of 680 and 850.

#### Step 2.1: Prime factorization of 680
To find the HCF, we can start with the prime factorization of 680:
\[ 680 = 2 \times 340 = 2 \times 2 \times 170 = 2 \times 2 \times 2 \times 85 = 2^3 \times 5 \times 17 \]

#### Step 2.2: Prime factorization of 850
Next, we factor 850:
\[ 850 = 2 \times 425 = 2 \times 5 \times 85 = 2 \times 5 \times 5 \times 17 = 2 \times 5^2 \times 17 \]

### Step 3: Identify common factors
Now we identify the common prime factors from the factorizations:
- For 680: \(2^3, 5, 17\)
- For 850: \(2, 5^2, 17\)

The common factors are:
- \(2^1\) (the minimum power of 2)
- \(5^1\) (the minimum power of 5)
- \(17^1\) (the minimum power of 17)

### Step 4: Calculate the HCF
Now we can calculate the HCF:
\[ \text{HCF} = 2^1 \times 5^1 \times 17^1 = 2 \times 5 \times 17 \]
Calculating this gives:
\[ 2 \times 5 = 10 \]
\[ 10 \times 17 = 170 \]

### Step 5: Find the product of the digits of the HCF
Now we need to find the product of the digits of the HCF, which is 170.
The digits of 170 are 1, 7, and 0.

Calculating the product:
\[ 1 \times 7 \times 0 = 0 \]

### Final Answer
Thus, the product of the digits of the greatest number that divides 690 and 875, leaving remainders of 10 and 25 respectively, is:
\[ \boxed{0} \]