When a certain number is divided by 65, the remainder is 56. When the same numberis divided by 13, the remainder is x . What is the value of `sqrt(5x-2)` ?

Options

A. `2sqrt(7)`
B. `sqrt(13)`
C. `2sqrt(2)`
D. `3sqrt(2)`

Answer

To solve the problem step by step, we need to find the value of \( x \) when a certain number leaves a remainder of 56 when divided by 65, and then find the value of \( \sqrt{5x - 2} \).

### Step 1: Understand the relationship between the number and the remainders
Let the certain number be \( N \). According to the problem, when \( N \) is divided by 65, the remainder is 56. This can be expressed mathematically as:
\[ N = 65k + 56 \]
for some integer \( k \).

### Step 2: Find the value of \( x \)
Next, we need to find the remainder when \( N \) is divided by 13. We can substitute \( N \) from the previous equation into this division:
\[ N = 65k + 56 \]

Now, we will calculate \( N \mod 13 \):
1. First, find \( 65 \mod 13 \):
\[ 65 \div 13 = 5 \quad \text{(exact division, remainder 0)} \]
So, \( 65 \equiv 0 \mod 13 \).

2. Next, find \( 56 \mod 13 \):
\[ 56 \div 13 = 4 \quad \text{(which gives a remainder of 4)} \]
So, \( 56 \equiv 4 \mod 13 \).

Putting it all together:
\[ N \mod 13 = (65k + 56) \mod 13 \]
\[ N \mod 13 \equiv (0 + 4) \mod 13 \]
Thus, \( N \mod 13 \equiv 4 \).

This means that when \( N \) is divided by 13, the remainder \( x \) is:
\[ x = 4 \]

### Step 3: Calculate \( \sqrt{5x - 2} \)
Now that we have \( x = 4 \), we can substitute this value into the expression \( \sqrt{5x - 2} \):
\[ \sqrt{5x - 2} = \sqrt{5(4) - 2} \]
\[ = \sqrt{20 - 2} \]
\[ = \sqrt{18} \]
\[ = \sqrt{9 \times 2} \]
\[ = 3\sqrt{2} \]

### Final Answer
Thus, the value of \( \sqrt{5x - 2} \) is:
\[ 3\sqrt{2} \]