Each question has four options(A), (B), (C) and (D) for answers. Select the right answer and write in English letters in the box against each question in the enclosed answer sheet.
A number, when divided by 296, gives 75 as the remainder. If the same number is divided by 37 then the remainder will be

Options

A. 1
B. 2
C. 19
D. 31

Answer

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

### Step 1: Understand the Problem
We are given that a number \( x \) when divided by 296 gives a remainder of 75. This can be expressed mathematically as:
\[ x = 296k + 75 \]
for some integer \( k \).

### Step 2: Find the Number Modulo 37
We need to find the remainder when the same number \( x \) is divided by 37. To do this, we can substitute the expression for \( x \) into the modulo operation:
\[ x \mod 37 = (296k + 75) \mod 37 \]

### Step 3: Simplify the Expression
We can separate the terms in the expression:
\[ x \mod 37 = (296k \mod 37 + 75 \mod 37) \mod 37 \]

### Step 4: Calculate \( 296 \mod 37 \)
Now we need to calculate \( 296 \mod 37 \):
1. Divide 296 by 37:
\[ 296 \div 37 \approx 8 \]
(since \( 37 \times 8 = 296 \))
2. Therefore, \( 296 \mod 37 = 0 \).

### Step 5: Calculate \( 75 \mod 37 \)
Next, we calculate \( 75 \mod 37 \):
1. Divide 75 by 37:
\[ 75 \div 37 \approx 2 \]
(since \( 37 \times 2 = 74 \))
2. The remainder is:
\[ 75 - 74 = 1 \]
3. Therefore, \( 75 \mod 37 = 1 \).

### Step 6: Combine the Results
Now we can substitute back into our expression:
\[ x \mod 37 = (0 + 1) \mod 37 = 1 \]

### Final Answer
Thus, the remainder when the number \( x \) is divided by 37 is:
\[ \text{Remainder} = 1 \]