If the number `670458x9595y` is divisible by `144`, then find the value of `sqrt(2x+3y)`

Options

A. `sqrt(11)`
B. `sqrt(13)`
C. `sqrt(12)`
D. 4

Answer

To solve the problem, we need to find the values of \( x \) and \( y \) such that the number \( 670458x9595y \) is divisible by \( 144 \). Since \( 144 = 9 \times 16 \), we will check the divisibility by both \( 9 \) and \( 16 \).

### Step 1: Check divisibility by 16
For a number to be divisible by \( 16 \), the last three digits must form a number that is divisible by \( 16 \). The last three digits of our number are \( 95y \).

We can express this as:
\[ 950 + y \]

Now we will check the values of \( y \) from \( 0 \) to \( 9 \) to find when \( 950 + y \) is divisible by \( 16 \).

Calculating \( 950 \mod 16 \):
\[ 950 \div 16 = 59.375 \quad \Rightarrow \quad 59 \times 16 = 944 \quad \Rightarrow \quad 950 - 944 = 6 \]
Thus, \( 950 \equiv 6 \mod 16 \).

To find \( y \) such that \( 950 + y \equiv 0 \mod 16 \):
\[ 6 + y \equiv 0 \mod 16 \quad \Rightarrow \quad y \equiv -6 \equiv 10 \mod 16 \]
Since \( y \) must be a single digit (0 to 9), we can check:
- If \( y = 2 \): \( 950 + 2 = 952 \) which is divisible by \( 16 \).

### Step 2: Check divisibility by 9
Next, we need to ensure that the entire number is divisible by \( 9 \). The rule for divisibility by \( 9 \) states that the sum of the digits must be divisible by \( 9 \).

Calculating the sum of the digits:
\[ 6 + 7 + 0 + 4 + 5 + 8 + x + 9 + 5 + 9 + 5 + y \]
This simplifies to:
\[ 6 + 7 + 0 + 4 + 5 + 8 + 9 + 5 + 9 + 5 + x + y = 58 + x + y \]

Substituting \( y = 2 \):
\[ 58 + x + 2 = 60 + x \]

Now, we need \( 60 + x \) to be divisible by \( 9 \):
Calculating \( 60 \mod 9 \):
\[ 60 \div 9 = 6.666 \quad \Rightarrow \quad 6 \times 9 = 54 \quad \Rightarrow \quad 60 - 54 = 6 \]
Thus, \( 60 \equiv 6 \mod 9 \).

To find \( x \):
\[ 6 + x \equiv 0 \mod 9 \quad \Rightarrow \quad x \equiv -6 \equiv 3 \mod 9 \]
Thus, \( x = 3 \) is the only valid single-digit solution.

### Step 3: Calculate \( \sqrt{2x + 3y} \)
Now that we have \( x = 3 \) and \( y = 2 \), we can substitute these values into the expression \( \sqrt{2x + 3y} \):
\[ \sqrt{2(3) + 3(2)} = \sqrt{6 + 6} = \sqrt{12} \]

### Final Answer
Thus, the value of \( \sqrt{2x + 3y} \) is \( \sqrt{12} \).