Might be a power problem. According to the [specification](http://www.starhub.com/content/dam/business/support/Internet/userguides/PDF/e3531-product-spec.pdf), E3531 consumes 500mA of current. Furthermore, "Maximum power consumption" is listed to be below 3W, which means you should expect peaks of up to 600mA, or even more the voltage on USB ports is below 5V. Since your RPi 3 provides 1.2A to its 4 USB ports, and assuming E3531 might use 600mA, you have another 600mA left for all other devices. You mention keyboard and mouse, which usually stay under 100mA, so if nothing else is connected you should be fine. If you have something else, or your keyboard and mouse are fancy (wireless, backlight etc.) you should check if you stay under the 1.2A limit. Alternatively, just plug the modem into a powered hub if you have one.

Since you mentioned soldering 5V line to USB ports of RPi, I invite you to be careful and check board schematics. RPi uses [this chip](http://www.diodes.com/_files/datasheets/AP255x.pdf) to provide current to USB ports, and it will most probably dislike it if you connect its output to 5V. Simple bypasses like [this one](https://raspberrypi.stackexchange.com/a/342/33476) worked on old RPi boards which had polyfuses.

Anyway, the problem you're having may not be power related at all. Check the output of `dmesg` for anything suspicious, especially for something repeated every 50 seconds.