After writing for the students where they have gone wrong with their given proofs, I realized that it was just me thinking they are wrong, just because I was accustomed to seeing the fundamental theorem of arithmetic explicitly used and stated. To see the point, let me quote the proofs given by J. L. Lagrange in his Lectures on Elementary Mathematics (1898) and More than half a century earlier (1831), Augustus De Morgan in his book ON THE STUDY AND DIFFICULTIES OF
MATHEMATICS (I've seen these in cut-the-knot)
Lagrange: it's impossible to find a whole number which multiplied by
itself will give 2. It cannot be found in fractions, for if you take a
fraction reduced to its lowest terms, the square of this fraction will
again be a fraction reduced to its lowest terms, and consequently
cannot be equal to the whole number 2.
Augustus De Morgan : 7 is not made by the multiplication either of any whole number or fraction by itself. The first is evident; the
second cannot be readily proved to the beginner, but he may, by taking
a number of instances, satisfy himself of this, that no fraction which
is really such, that is whose numerator is not measured by its
denominator, will give a whole number when multiplied by itself, thus
4/3×4/3=16/9 is not a whole number, and so on.
True, both of them appeal to the fundamental theorem of Arithmetic indirectly. But, both of them are based on what the authors have assumed to be correct previously. This reminds me of the role of the context of a proof as described by Ian Stewart and David Tall in The Foundations of Mathematics
One situation that is of great concern to students is what constitutes
a proof acceptable in an examination. To a certain extent, the answer
depends on the examiner, but part of the anxiety is due to uncertainty
about the context...in an examination it may not be clear at which
level a proof is required. Do all the steps have to be included? What
can safely be missed out?
Thus, it seems that I should change all the comments I have written for the students, and instead, ask them to specify where they have used the fundamental theorem of arithmetic! And in the future, If the use of a certain tool/concept/approach matters to me, I should specify it at the outset!!