Your example is too crude to provide anything to correct.
First of all if you are mining for a pool you really aren't a "miner" you are a SHA-2 hashpower provider (independent contractor), your pool makes sure there is no duplicate work. For the purpose of this explanation, only entities which construct the block headers are "miners" (pool operators, solo miners, and some unique situations like p2pool).
No miner is working on the same "work". You aren't just hashing a single number. That would be beyond pointless it wouldn't provide any security. What does the number 123456 hashed mean? Absolutely nothing. How would it prevent modifying the blockchain? It wouldn't.
You are hashing a blockheader which consists of 6 values only one of which is "the number" which is called a nonce. The nonce is simply a way to add entropy so you can make multiple tries without changing anything else in the blockheader.
So what is in the blockheader?
Field Purpose Updated when
Version Block version number You upgrade the software and it specifies a new version
hashPrevBlock 256-bit hash of the previous block header A new block comes in
hashMerkleRoot 256-bit hash based on all of the transactions in the block A transaction is accepted
Time Current timestamp as seconds since 1970-01-01T00:00 UTC Every few seconds
Bits Current target in compact format The difficulty is adjusted
Nonce 32-bit number (starts at 0) A hash is tried (increments)
https://en.bitcoin.it/wiki/Block_hashing_algorithmIf ANY of these values are different the hash will be different. Miners routinely attempt the same nonce. Your own individual miners will attempt the same nonce millions of times. The nonce is only 32 bit, there are only 4 billion possibilities. A KNC Jupiter for example will exhaust the nonce range in less than 1/100th of a second. Even a low end GPU will try all possible nonces (for a given blockheader) in a few seconds.
The "hashMerkleRoot" is a unique hash which represents all the transactions in the block.
If you add, remove, or change ONE transaction in the block you will end up with a different Merkle Root hash. One of those transactions is the "coinbase" where miners receive their reward. Even two miners (solo miners, pools, p2pool users) which have the exact same set of other transactions will still have a unique coinbase txs and thus have different Merkle Root hashes.
Different Merkle Root Hash = Different Block Header = Different Hash.
No two miners on the planet are ever working on the same work (with the exception of a bug in a pool server issuing duplicate work to its workers).