Is 43 a prime number?

Yes
A prime number is one that has as its factors only itself and 1. So for instance 3 is a prime number and 4 is not:
##1xx3=3## and that’s all we can get for 3 so 3 is prime
##1xx4=4, 2xx2=4## – 4 has two sets of factors and so isn’t prime
So are there any factors for 43 other than ##43xx1##?
The quick answer is “no” and so 43 is prime.
The longer answer is that we can try to multiply numbers to get to 43:
43 is odd and so nothing times 2 will work
##4+3=7## which is not divisible by 3, so nothing times 3 will work
43 doesn’t end in 0 or 5 so nothing times 5 will work
##7xx6=42##. Close but no cigar – nothing times 7 will work
##11xx4=44##. Close again but still no – so nothing times 11 will work
##13xx3=39##. Not so close and still doesn’t work – so nothing times 13 will work
##17xx2=34, 17xx3=51##. Not close at all and still doesn’t work – so nothing times 17 will work.
##19xx2=38, 19xx3=57##. Nope – nothing times 19 works
##23xx2=46##. Nope.
And there’s no sense in continuing – if 2 times 23 is too big, nothing beyond this point will be smaller. And, in fact, we could have stopped much earlier because we already knew anything times 2, 3, 4, 5,… wouldn’t work (I think technically the last attempt that was redundant was seeing that 5 was not a factor – but it was good to show definitively that these bigger numbers wouldn’t work).