Why even divided by odd equal even?

Answer 1

Odd numbers do NOT have a factor of 2.

When an Even number is divided by an Odd number with no remainder, the prime factors of the divisor cancel out the prime factors of the dividend, but there will be no prime 2 to cancel the prime factor of 2 in the dividend leaving the the quotient to have a factor of 2, and so will be Even.

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Alternatively, consider what happens when you multiply Odd and Even numbers:

Odd × Odd = Odd

Odd × Even = Even

Even × Even = Even

The result is Even ONLY if at least one of the numbers multiplied together is Even.

Thus if you divide an Even number by an Odd number, the result must be Even.

Also consider that Odd times Odd is Odd. Multiplication is the inverse of division, so if you have:

dividend ÷ divisor = quotient

then

dividend = quotient × divisor

Now, if the divisor is Odd and the quotient is Odd, the the Dividend MUST be Odd (since Odd × Odd = Odd).

Thus if the Dividend is Even and the Divisor is Odd, then the Quotient must be Even.