Use RSA private key to generate public key?

I don't really understand this one:

According to https://www.madboa.com/geek/openssl/#key-rsa, you can generate a public key from a private key.

openssl genrsa -out mykey.pem 1024
openssl rsa -in mykey.pem -pubout > mykey.pub

My initial thinking was that they are generated in a pair together.

Does the RSA private key contain the sum? Or the public key?

openssl genrsa -out mykey.pem 1024

will actually produce a public - private key pair. The pair is stored in the generated mykey.pem file.

openssl rsa -in mykey.pem -pubout > mykey.pub

will extract the public key and print that out. Here is a link to a page that describes this better.

EDIT: Check the examples section here. To just output the public part of a private key:

openssl rsa -in key.pem -pubout -out pubkey.pem

To get a usable public key for SSH purposes, use ssh-keygen:

ssh-keygen -y -f key.pem > key.pub

People looking for SSH public key...

If you're looking to extract the public key for use with OpenSSH, you will need to get the public key a bit differently

$ ssh-keygen -y -f mykey.pem > mykey.pub

This public key format is compatible with OpenSSH. Append the public key to remote:~/.ssh/authorized_keys and you'll be good to go

docs from SSH-KEYGEN(1)

ssh-keygen -y [-f input_keyfile]  

-y This option will read a private OpenSSH format file and print an OpenSSH public key to stdout.

In most software that generates RSA private keys, including OpenSSL's, the private key is represented as a PKCS#1 RSAPrivatekey object or some variant thereof:

A.1.2 RSA private key syntax

An RSA private key should be represented with the ASN.1 type
RSAPrivateKey:

  RSAPrivateKey ::= SEQUENCE {
      version           Version,
      modulus           INTEGER,  -- n
      publicExponent    INTEGER,  -- e
      privateExponent   INTEGER,  -- d
      prime1            INTEGER,  -- p
      prime2            INTEGER,  -- q
      exponent1         INTEGER,  -- d mod (p-1)
      exponent2         INTEGER,  -- d mod (q-1)
      coefficient       INTEGER,  -- (inverse of q) mod p
      otherPrimeInfos   OtherPrimeInfos OPTIONAL
  }

As you can see, this format has a number of fields including the modulus and public exponent and thus is a strict superset of the information in an RSA public key.