Find the smallest number which us divided by 8 & 5 and leaves remainder 4 and 3 but it is exactly divided by 13.

Options

A. 578
B. 478
C. 563
D. 468

Answer

To find the smallest number that, when divided by 8 and 5, leaves remainders of 4 and 3 respectively, and is exactly divisible by 13, we can follow these steps:

### Step 1: Set Up the Equations
We know that:
- The number \( N \) leaves a remainder of 4 when divided by 8. This can be expressed as:
\[ N = 8a + 4 \]
where \( a \) is an integer.

- The number \( N \) leaves a remainder of 3 when divided by 5. This can be expressed as:
\[ N = 5b + 3 \]
where \( b \) is an integer.

### Step 2: Equate the Two Expressions
Since both expressions represent the same number \( N \), we can set them equal to each other:
\[ 8a + 4 = 5b + 3 \]
Rearranging gives us:
\[ 5b - 8a = 1 \]

### Step 3: Solve for \( b \)
From the equation \( 5b = 8a + 1 \), we can express \( b \) as:
\[ b = \frac{8a + 1}{5} \]
For \( b \) to be an integer, \( 8a + 1 \) must be divisible by 5.

### Step 4: Find Suitable Values of \( a \)
We will test integer values of \( a \) to find when \( b \) is an integer:
- For \( a = 0 \): \( b = \frac{1}{5} \) (not an integer)
- For \( a = 1 \): \( b = \frac{9}{5} \) (not an integer)
- For \( a = 2 \): \( b = \frac{17}{5} \) (not an integer)
- For \( a = 3 \): \( b = \frac{25}{5} = 5 \) (integer)

### Step 5: Calculate \( N \)
Now that we have \( a = 3 \) and \( b = 5 \), we can find \( N \):
\[ N = 8a + 4 = 8(3) + 4 = 24 + 4 = 28 \]

### Step 6: Check Divisibility by 13
We need to ensure that \( N \) is also divisible by 13. Since \( 28 \) is not divisible by 13, we need to find the smallest number of the form \( 28 + 40k \) (where \( 40 \) is the LCM of 8 and 5) that is divisible by 13.

### Step 7: Set Up the Divisibility Condition
We want:
\[ 28 + 40k \equiv 0 \mod{13} \]
Calculating \( 28 \mod 13 \):
\[ 28 \equiv 2 \mod{13} \]
So we need:
\[ 2 + 40k \equiv 0 \mod{13} \]
Calculating \( 40 \mod 13 \):
\[ 40 \equiv 1 \mod{13} \]
Thus, we have:
\[ 2 + k \equiv 0 \mod{13} \]
This simplifies to:
\[ k \equiv -2 \equiv 11 \mod{13} \]

### Step 8: Find the Smallest \( k \)
The smallest positive integer \( k \) that satisfies this is \( k = 11 \).

### Step 9: Calculate the Final Value of \( N \)
Now substituting \( k = 11 \) back into our equation for \( N \):
\[ N = 28 + 40(11) = 28 + 440 = 468 \]

### Conclusion
The smallest number that meets all the conditions is:
\[ \boxed{468} \]