How can I convert an input with the 16-bit Sigma delta A/D converter
In line with the site policy about homework-type questions (meaning that we give socratic questions and hints, rather than just stating the full answer), here are some things to consider, to lead you towards the correct answer from your given list.
FYI, some helpful details aren't in the PIC24 Family Datasheet (which has very little about that ADC) but instead are in the PIC24 Family Reference Manual (FRM). The relevant section from that FRM is available here (linked from the main PIC24FJ128GC010 product page).

Think about what it means, that it's a 16-bit ADC, rather than 12-bit or 10-bit etc. (See the comment from hobbs above.) What digital value represents the maximum analog input voltage (before saturation)?

On that ADC, what would be the maximum analog input voltage which could be converted (whilst avoiding saturation) if GAIN = 1? As you know, it's the voltage between two specific pins. There are two possible voltages in the remaining answer choices.

How does GAIN = 2 affect the maximum analog input voltage which can then be represented in the maximum digital output value, i.e. how does the Vin range change? See the extract from the FRM which I linked, specifically section 66.3.2 &quot;Gain Control&quot;.

If you are comfortable thinking in hexadecimal, look at the ADC output values in the remaining possible answer choices (they are 0x7AE0 and 0x3D70) and compare them to the maximum ADC output value as approximate fractions. (If it helps you, you can convert everything to decimal.) What can you say about the approximate values given in the remaining answer choices, as fractions of the maximum?
If you can get this far, then you don't even need a calculator to find the right answer choice - but it's no shame if you prefer to go through it with a calculator.


If you can make some conclusions from those steps, then you are close to finding which one of the multiple-choice answers is correct.