S1::

Suppose that Boris and Natasha brought in x 50 ruble notes and y 100 ruble notes. Their sum is thus

50x + 100y

Since x and y are nonnegative integers, this sum MUST be even. (We have Even + Even, which is always Even.)

Now let's look at the change they received. Suppose they got a, b, c, and 1, 3, 5, and 25 ruble notes, respectively. Their change is

1a + 3b + 5c + 25d

Looks fine, right? But here's the trick! If we have an EVEN number of odd numbers, our sum will be even. For instance:

1 + 3 + 5 + 7 = 16
9 + 17 + 5 + 11 = 52
etc.

But if we have an ODD number of odd numbers, our sum will be odd.

1 + 3 + 5 = 9
7 + 11 + 23 + 1 + 1 = 43
etc.

Since the teller gives Boris and Natasha 2013 notes, he is giving them an ODD number of bills, each of which has an ODD value. As we saw above (or remember from our basic number properties), an odd number of odd numbers has an odd sum. But Boris and Natasha's sum must have been even! It was 50x + 100y.

So they cannot have been given the correct change, and S1 is SUFFICIENT.

S2:: On its own, not very helpful.