A number, when divided by 136, leaves remainder 36. If the same number is divided by 17, the remainder will be

Options

A. 9
B. 7
C. 3
D. 2

Answer

To solve the problem step by step, we follow these instructions:

### Step 1: Understand the problem
We are given that a number leaves a remainder of 36 when divided by 136. We need to find the remainder when the same number is divided by 17.

### Step 2: Express the number mathematically
Let the number be represented as \( N \). According to the problem, we can express \( N \) using the formula for division:
\[ N = 136k + 36 \]
where \( k \) is some integer (the quotient when \( N \) is divided by 136).

### Step 3: Simplify the expression
We can simplify the expression for \( N \):
\[ N = 136k + 36 \]

### Step 4: Find \( N \) modulo 17
To find the remainder when \( N \) is divided by 17, we can take the expression we have for \( N \) and find it modulo 17:
\[ N \mod 17 = (136k + 36) \mod 17 \]

### Step 5: Calculate \( 136 \mod 17 \) and \( 36 \mod 17 \)
First, we need to find \( 136 \mod 17 \):
\[ 136 \div 17 = 8 \quad \text{(since } 17 \times 8 = 136\text{)} \]
Thus,
\[ 136 \mod 17 = 0 \]

Next, we calculate \( 36 \mod 17 \):
\[ 36 \div 17 = 2 \quad \text{(since } 17 \times 2 = 34\text{)} \]
Thus,
\[ 36 \mod 17 = 36 - 34 = 2 \]

### Step 6: Substitute back into the equation
Now substituting back into our equation:
\[ N \mod 17 = (0 \cdot k + 2) \mod 17 = 2 \]

### Step 7: Conclusion
Thus, the remainder when the number \( N \) is divided by 17 is:
\[ \text{Remainder} = 2 \]