A number when divided by296 leaves 75 as remainder .when the same number is divided by 37,the remainder will be?

To solve this problem, let's denote the unknown number by \( x \).

### Given Information:
1. When \( x \) is divided by 296, it leaves a remainder of 75. This can be expressed mathematically as:
   \[
   x = 296k + 75
   \]
   where \( k \) is some integer.

2. We want to find the remainder when the same number \( x \) is divided by 37.

### Step-by-Step Solution:

First, substitute the expression for \( x \) into the division by 37:

\[
x = 296k + 75
\]

Now, we need to find \( x \mod 37 \). 

Since \( x = 296k + 75 \), we can write:

\[
x \mod 37 = (296k + 75) \mod 37
\]

Using the property of modulus:

\[
x \mod 37 = [(296k) \mod 37 + (75) \mod 37] \mod 37
\]

Next, calculate each term separately:
1. **Find \( 296 \mod 37 \):**
   \[
   296 \div 37 = 8 \text{ remainder } 0 \implies 296 \equiv 0 \pmod{37}
   \]

   So, \( 296k \mod 37 = 0 \).

2. **Find \( 75 \mod 37 \):**
   \[
   75 \div 37 = 2 \text{ remainder } 1 \implies 75 \equiv 1 \pmod{37}
   \]

So, 

\[
x \mod 37 = (0 + 1) \mod 37 = 1
\]

### Conclusion:

When the number is divided by 37, the remainder will be **1**.

Would you like further details or have any questions?

Here are some related questions to explore further:

1. How do you solve similar problems with different divisors?
2. How does the modulus operation work for negative numbers?
3. What is the significance of modulus in number theory?
4. Can this approach be used to solve equations involving multiple congruences?
5. How would this problem change if both divisors were prime numbers?

**Tip:** When working with remainders, the modulus operation is a powerful tool to simplify the expressions and find solutions efficiently.

Learn how to find the remainder when a number, which leaves a remainder of 75 when divided by 296, is divided by 37. This problem involves modular arithmetic and number theory concepts, specifically the modulus operation. Suitable for students in Grades 9-12.

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