Data Sufficiency

In the former Soviet Union, rubles came in denominations of 1, 3, 5, 10, 25, 50, and 100. Boris and Natasha are at the bank to change an enormous pile of 50 and 100 ruble notes, but the teller is out of 10s and is forced to give them their change in nothing but 1, 3, 5, and 25 ruble notes. Did the teller give Boris and Natasha the correct change?

(1) The teller gave Boris and Natasha 2013 notes.
(2) The teller did not give Boris and Natasha any 3 ruble notes.

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.