Q: Is zero a never or sometimes or a always divisor?

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Never.

Zero is never a divisor. If you ever see a fraction or a problem where zero is adivisor, you can stop right there and toss the whole thing. In the language ofmath, "Division by zero is not permitted".

The sum of zero and any integer is never zero.And it's still 'integer', not 'interger'.

Division by an integer is always defined only when the divisor is not zero

The opposite of zero - in the sense of additive inverse - is zero.

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Never.

Zero is never a divisor. If you ever see a fraction or a problem where zero is adivisor, you can stop right there and toss the whole thing. In the language ofmath, "Division by zero is not permitted".

The sum of zero and any integer is never zero.And it's still 'integer', not 'interger'.

It is a rational number.

Division by an integer is always defined only when the divisor is not zero

The sum of zero and a negative integer can never be zero - it will always be negative and nonzero. Although zero is also an integer, it is neither negative nor positive and cannot be the other integer used.

Sometimes. The opposite of zero depends on the type of function under consideration. For example, the additive opposite of zero is zero. The multiplicative opposite is not defined.

It is always true.

The opposite of zero - in the sense of additive inverse - is zero.

For zero to be a factor of a number, there would have to be another factor paired with it. Since zero times anything is zero, you will never be able to multiply zero with anything to get any number other than zero.

The possible number of remainders is always one less than the divisor.

Division can be thought of as the opposite of multiplication: 0 ÷ 3 is the same as saying "what number when multiplied by 3 results in 0"; answer: 0. 3 ÷ 0 is the same as saying "what number when multiplied by 0 results in 3"; no number when multiplied by 0 results in 3 (as 0 times anything is 0), thus it can't be done. Alternatively, division tells you how many times you need to, or can, subtract the divisor from the dividend to get to zero. If you start with a dividend of zero and a non-zero divisor, you don't need to, nor can you, subtract the divisor to get to zero. If you start with a non-zero dividend, and a zero divisor, no matter how many times you subtract the divisor you will never get to zero - the dividend stays the same. With a zero dividend and a zero divisor, you have reached zero when you start, BUT you can subtract the divisor and the dividend will then become (stay) zero; thus zero divided by zero is any number you want - in calculus there are rules which specify the value to use in different circumstances.