I know when subtracting or adding significant figures,it is necessary to round the resultant to the least amount of significant figures, on the right sight of the decimal point, based on the least amount of significant figures of a number involved in the sum or resultant. I want to clarify now, is there ever a case involving the division or multiplication of significant figures, where it would be necessary to round the product or quotient?
It's a matter of carrying the accuracy through to your answer. The number of significant figures in the inputs identifes their resolution (this is different from physical accuracy).
For example 1.234 has 4 siginificant figures and the actual figure could be anywhere between 1.2335 and 1.2345, i.e. it has a resolution of 0.001. Having determined the resolution, it needs to be propagated through the division/multiplication or other operation correctly (not covered here).
Any figures smaller than the propagated resolution are not significant in the result.