The various other prime numbers are all odd number such together \$5, 11, 127,\$ and \$37\$. So, why is \$2\$ the just prime even number over there is?

Is it since it only has actually 1 and itself the way, even though it"s even? check it out on this excellent aramuseum.org page one-fourth native the bottom.

You are watching: Can an even number be prime

Why is 2 the only also \$(\$binary\$)\$ number that is prime?

For the same reason that three is the just ternary number the is prime. I beg your pardon is the same reason for which 5 is the just quinary number the is prime. Etc.

Because every also number other than 2 is obviously divisible through 2 and also so by definition cannot be prime.

A (positive) also number is some number \$n\$ such that \$n = 2 cdot k\$ for some (positive) integer \$k\$. A element number has actually only itself and also \$1\$ together (positive) divisors.

What wake up if \$n ot = 2\$ in our definition of also numbers?

Pick a element \$p\$. Contact a number \$n\$ \$p\$-divisible if \$pmid n\$. Climate \$p\$ is the just \$p\$-divisible prime, trivially. In particular, \$2\$ is the just \$2\$-divisible, or even, prime.

In the integers, \$-2\$ is another even prime.

For variety, in the Gaussian integers, \$2\$ is not prime: e.g. Components as \$(1+i)(1-i)\$. The also primes that the Gaussian integers are \$pm 1 pm i\$, return these are all the "same" element in the very same sense the in the integers, \$2\$ and \$-2\$ are the "same" prime.

(I specify "even" in a number field to be identical to its share being even)

In the ring of every rational numbers with odd denominator, \$2/7\$ is an also prime. In fact, \$2/n\$ is prime for every odd creature \$n\$. (but again, these room all the "same" prime)

There are also number rings that have distinctive even primes that room not the "same" in the feeling implied above.

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reply Nov 3 "14 at 2:00
user14972user14972
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A element number is such the it is divisible by only itself and also one. Consisting of 1 as a prime number would certainly violate the basic theory of arithmetic, for this reason in modern aramuseum.orgematics the is excluded. Two is a prime since it is divisible by just two and one. All the other even numbers are not prime due to the fact that they are all divisible by two. That leaves just the strange numbers. Of course, not all strange numbers are prime (e.g. Nine is divisible by three).

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answered january 28 "15 in ~ 8:36

Michael LeeMichael Lee
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The word prime comes from the Latin indigenous primus which method "first." two (2) is the first even number. In various other words, it starts all the even numbers. There is an ext than one odd prime number since odd numbers are never ever divisible by 2.

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answered Mar 26 "17 at 0:32
Carly Brooke SteffenCarly Brooke Steffen
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The number \$2\$ has actually only two whole number factors, \$1\$ and also itself. That"s pretty lot it ~ this: The other numbers the are even up indigenous two room all divisible by the number in part way. This is also known together the "oddest prime" since it"s the only prime number that"s even, so it"s additionally known as the odd one out. Ns guess now that that"s pretty much it going come the question about why this have the right to happen.

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reply Nov 3 "14 in ~ 1:45
aramuseum.orgsteraramuseum.orgster
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