IP address, changed by router?

I was talking to a friend today and he told me that a router changes the IP address you're given to by the ISP?
So I checked with whatismyip.org and let's say with the router, it showed, 29.28.29.450
Then I took the router out and just used the modem and it showed as 29.28.29.449

If I were to use a different router, would it end with 450 again? Or an entirely different number, because the router is of a different brand?

I'm assuming having a router is good for protection from malware, trojans and what not?

Thanks.
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More about address changed router
  1. It depends on how your ISP handles IP addresses. Some use DHCP so every time you disconnect then reconnect you get a new address that happens to be free. Some have smarter DHCP servers like my ISP that remembers your device's mac address (unique identifier assigned to network interfaces) So I would get a different ip address on each device it plugged directly in to my modem, but if I plugged a device back in again it would get the same IP it had before.

    But I would recommend using a router, it blocks allot more stuff trying to break in to your network, as well as adds the ability let multiple device connect to the internet at once
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