Determine input resistance of a DMM using a voltage follower
When you're measuring the voltage direct with the DMM you basically have this arrangement:







simulate this circuit &ndash; Schematic created using CircuitLab

From that you can calculate what the total resistance in the lower half must be in order to get 5V at the junction. 

When you have calculated that total resistance, you can then separate out the two resistors in parallel to work out what R3 must equate to.