The largest remainder when dividing by 9 is 8.
When performing division, the remainder is the amount left over after one whole number is divided by another, resulting in a whole quotient and a remainder that is less than the divisor. In the context of dividing by 9, the possible remainders can range from 0 up to 8.
Understanding Remainders
A remainder is what's left over when a number cannot be perfectly divided by another. For any division problem in the form:
- Dividend ÷ Divisor = Quotient with a Remainder
The remainder must always be a non-negative integer and less than the divisor. If the remainder were equal to or greater than the divisor, it would mean that another full multiple of the divisor could have been extracted from the original number, leading to an increased quotient and a smaller (or zero) remainder.
For instance, if you divide a number by 9:
- A remainder of 9 is not possible because if you have 9 left over, that means another group of 9 could have been taken out, making the remainder 0 and the quotient one larger.
- Similarly, a remainder of 10 would mean another group of 9 could be taken out, leaving a remainder of 1.
Therefore, the largest possible remainder when dividing by 9 is always 8. This rule applies universally: the maximum remainder for any division is always one less than the divisor.
Practical Examples of Division by 9
Let's look at some examples to illustrate how remainders work when dividing by 9:
| Dividend | Divisor | Quotient | Remainder | Calculation |
|---|---|---|---|---|
| 17 | 9 | 1 | 8 | 17 = (1 × 9) + 8 |
| 26 | 9 | 2 | 8 | 26 = (2 × 9) + 8 |
| 35 | 9 | 3 | 8 | 35 = (3 × 9) + 8 |
| 9 | 9 | 1 | 0 | 9 = (1 × 9) + 0 |
| 18 | 9 | 2 | 0 | 18 = (2 × 9) + 0 |
| 10 | 9 | 1 | 1 | 10 = (1 × 9) + 1 |
As you can see from the examples, no matter what number you divide by 9, the remainder will always be a whole number between 0 and 8, inclusive. The value 8 represents the largest possible "leftover" before another full group of 9 could be formed.
For more information on the concept of remainders and division, you can explore resources like Wikipedia's article on Euclidean division.
