A deck of 100 cards is numbered from 1 to 100. Each card has the same number printed on both sides. One side of each card is red and the other side is yellow. Barsby places all the cards, red side up, on a table. He first turns over every card that has a number divisible by 2 . He then examines all the cards, and turns over every card that has a number divisible by 3 . How many cards have the red side up when Barsby is finished?
(A) 83
(B) 17
(C) 66
(D) 50
(E) 49
A deck of 100 cards is numbered from 1 to 100 . Each card has the same number printed on both sides. One side of each card is red and the other side is yellow. Barsby places all the cards, red side up, on a table. He first turns over every card that has a number divisible by 2 . He then examines all the cards, and turns over every card that has a number divisible by 3 . How many cards have the red side up when Barsby is finished?
(A) 83
(B) 17
(C) 66
(D) 50
(E) 49

## Solution

Initially, all 100 cards have the red side up. After Barsby's first pass only the 50 odd-numbered cards have the red side up, since he has just turned all the even-numbered cards from red to yellow.

During Barsby's second pass he turns over all cards whose number is divisible by 3. On this pass Barsby will turn any odd-numbered card divisible by 3 from red to yellow. Between 1 and 100, there are 17 odd numbers that are divisible by 3 , namely $3,9,15,21, \ldots, 93$, and 99 . Also on this pass, Barsby will turn any even-numbered card divisible by 3 from yellow to red. Between 1 and 100, there are 16 even numbers that are divisible by 3 , namely $6,12,18,24, \ldots, 90$, and 96 .

When Barsby is finished, the cards that have the red side up are the 50 odd-numbered cards from the first pass, minus the 17 odd-numbered cards divisible by 3 from the second pass, plus the 16 evennumbered cards divisible by 3 , also from the second pass.

Thus, $50-17+16=49$ cards have the red side up.

ANSWER: (E)