Question

remainders quick check

Answer 1

Sure! A remainder is what is left over after division when one number is not evenly divisible by another. Here’s a quick overview of how to find remainders:

1. **Basic Division**: When you divide two numbers, the quotient is the whole number result, and the remainder is what’s left over.
- Example: When you divide 10 by 3, you get 3 as the quotient (because 3 × 3 = 9), and the remainder is 10 - 9 = 1. So, 10 ÷ 3 = 3 R1.

2. **Formula**: The relationship can be expressed as:
\[
a = b \cdot q + r
\]
Where:
- \( a \) is the dividend
- \( b \) is the divisor
- \( q \) is the quotient
- \( r \) is the remainder and must be less than \( b \).

3. **Remainder with Modulus**: The remainder can also be found using the modulus operation.
For example:
- \( 10 \mod 3 = 1 \)

4. **Examples**:
- **Example 1**: 17 divided by 5
- \( 17 \div 5 = 3 \) (quotient), \( 5 \cdot 3 = 15 \), \( 17 - 15 = 2 \). **Remainder = 2**
- **Example 2**: 29 divided by 4
- \( 29 \div 4 = 7 \), \( 4 \cdot 7 = 28 \), \( 29 - 28 = 1 \). **Remainder = 1**

5. **Tips for Large Numbers**:
- Sometimes breaking down the division can help.
- For example, to find \( 123 \mod 11 \), you can first find how many times 11 fits into 123.

6. **Common Remainders**:
- Any number divided by itself has a remainder of 0.
- Any number divided by 1 has a remainder of 0.

If you have specific cases you'd like to check, feel free to share!