The problem from class 12 asked you show that Euler's phi-function was multiplicative in two examples and not multiplicative in a third example. Here "multiplicative" means
The question mark over the "equals" sign emphasizes that this statement is dubious. It seems to be true sometimes and not others. It is true, for example, that phi(35)=phi(5)*phi(7), and that phi(36)=phi(4)*phi(9), but it is NOT true that phi(9)=phi(3)*phi(3). That was the content of the problem from class 12.
The challenge problem is to clarify this situation. You can try more examples, make conjectures, test them, etc. An acceptable answer will show some coherent thought, and make some progress in clarifying what is going on. It need not actually prove anything, but it should organize strong evidence in favor of some idea or ideas of your own.
In class 19 we saw a method to decide if a number a is a square mod p where p is prime -- it amounts to computing the Legendre symbol by a certain algorithm, using quadratic reciprocity, etc., to see if it is 1 or -1. (By the way, if a is 0 mod p, the Legendre symbol is defined to be 0.) If you ask the same question mod n, however, where n is not prime, most of the algorithm doesn't apply. Quadratic reciprocity is a statement only about primes, etc. So this problem is to investigate which numbers a are squares mod n where n is, for simplicity, p*q, the product of two distinct primes. The first thing to do is probably just experiment. Since we also have a way of understanding multiplication in Zn* by the homomorphisms onto Zp* and Zq*, you might keep this in mind as you experiment. The problem is to gain an understanding of this case also.
This may be a rather silly problem, but I can't resist. You know from class 3 that the Egyptian representation for fractions required that the numerator be 1. A fraction like 1/7 was perfectly OK, for example. But the fraction we would represent as 2/7 was, for them, 1/4+1/28. That is, they wouldn't even write 1/7+1/7, but rather required the denominators to be different. This is obviously pretty clumsy, but my question is, does it even work? That is, can all fractions be written as a sum of fractions (perhaps more than just two fractions) with numerators equal to one, and denominators all different? A few examples will not be enough -- there must be some kind of general argument to show it either is or is not possible.