Each of the following 15 cards has a letter on one side and a positive integer on the other side.

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-022.jpg?height=371&width=523&top_left_y=1506&top_left_x=758)

What is the minimum number of cards that need to be turned over to check if the following statement is true?

"If a card has a lower case letter on one side, then it has an odd integer on the other side."
(A) 11
(B) 9
(C) 7
(D) 5
(E) 3
Each card fits into exactly one of the following categories:

(A) lower case letter on one side, even integer on the other side

(B) lower case letter on one side, odd integer on the other side

(C) upper case letter on one side, even integer on the other side

(D) upper case letter on one side, odd integer on the other side

The given statement is

"If a card has a lower case letter on one side, then it has an odd integer on the other."

If a card fits into category (B), (C) or (D), it does not violate the given statement, and so the given statement is true. If a card fits into category (A), it does violate the given statement. Therefore, we need to turn over any card that might be in category (A). Of the given cards,

(i) 1 card shows a lower case letter and might be in (A),

(ii) 4 cards show an upper case letter and is not in (A),

(iii) 2 cards show an even integer and might be in (A), and

(iv) 8 cards show an odd integer and is not in (A).

In order to check if this statement is true, we must turn over the cards in (i) and (iii), of which there are 3 .