All Primes Are Odd

Number Theory Level 1

True or False:

$\quad$ Since every even number is a multiple of 2, hence no even number is a prime .

True False

2 solutions

Zach Abueg
Jan 30, 2017

The counterexample is $2$ , which, although clearly a multiple of $2$ , has no other factors but itself and $1 -$ satisfying the definition of a prime number.

What is the error made in the logic presented in the problem?

Calvin Lin Staff - 4 years, 4 months ago

It misuses the definition of a prime number , saying that all multiples of $2$ are not prime.

Zach Abueg - 4 years, 4 months ago

In a certain sense, yes.

But, how could you explain that clearly to someone who doesn't understand it as yet? For someone who said "true", can you point out the error that they made in using statement 1 to conclude that statement 2 is true?

Calvin Lin Staff - 4 years, 4 months ago

This is closely related to the question of whether or not a prime gap of 1 is possible for numbers greater than 2.

Steven Chase - 4 years, 4 months ago

Calvin Lin Staff
Jan 31, 2017

For those who answered True, the misconception is in thinking "If a number has a factor that isn't 1, then the number cannot be prime." IE In this case, thinking that "Since 2 is a factor of the even number, hence this even number cannot be prime."

However, this condition isn't sufficient. Remember that a prime number has 2 factors, 1 and itself. So, to show that a number is not prime by finding factors, we have to show that is has at least 3 positive, distinct factors. Simply showing that there are 2 factors is not enough.

The counterexample to the statement is the number 2 itself, which is prime. Hence, this statement is false.

All Primes Are Odd

2 solutions

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