When any number is divided by 1, the remainder is always 0.
Division is a fundamental arithmetic operation that helps us determine how many times one number (the divisor) is contained within another number (the dividend), and what is left over. When you divide any integer by 1, you are essentially asking how many times the number 1 fits into that integer.
Since 1 is the multiplicative identity, any number multiplied by 1 remains itself. Similarly, any number divided by 1 is itself. This means that 1 fits into any number (N) exactly N times, leaving nothing behind. Therefore, the remainder will always be zero.
The division algorithm states:
Dividend = Quotient × Divisor + Remainder
When the divisor is 1, this equation becomes:
Number = Number × 1 + 0
For instance, if you have 10 items and you divide them into groups of 1 item each, you will have 10 groups, and there will be no items left over.
Examples of Division by 1
The following table illustrates the consistent result when dividing various numbers by 1:
| Dividend (N) | Divisor (D) | Quotient (Q) | Remainder (R) |
|---|---|---|---|
| 5 | 1 | 5 | 0 |
| 20 | 1 | 20 | 0 |
| 1 | 1 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| -7 | 1 | -7 | 0 |
Distinguishing from Division of One by Another Number
It's crucial to understand the difference between dividing a number by 1 (the question asked) and dividing 1 by another number. A common misconception can arise from scenarios like the one described in an external reference:
"The quotient is 0 and since 1 cannot be evenly divided by a number larger than itself, the remainder is simply 1. This is why 1 divided by any number (except itself) leaves a remainder of 1." — Why does 1 have a remainder of 1 when divided by any number?
This statement, from the provided source, refers to the operation where 1 is the dividend and it's being divided by a number larger than itself (e.g., 1 divided by 2, 1 divided by 5). In such cases, since 1 does not contain the larger number even once, the quotient is 0, and the entire 1 is left over as the remainder.
However, the original question asks "When a number is divided by 1," meaning 1 is the divisor. In this specific context, regardless of the integer being divided (the dividend), the remainder will always be 0 because 1 divides evenly into every integer.
