Back to Math Problems (Ages 12 to 13)
A deck of 90 cards is numbered from 1 to 90. Each card has a blue side and an orange side, with the number printed on both sides.
Alice arranges all the cards on a table with their blue sides facing up. She then follows these steps:
1. Flip over every card with a number that is a multiple of 3.
2. Flip over every card with a number that is a multiple of 4.
3. Flip over every card with a number that is a multiple of 5.
After Alice has completed all these steps, how many cards have their orange side facing up?
Let’s approach this step-by-step:
1. First, let’s consider which cards are flipped in each step:
– Multiples of 3: 3, 6, 9, 12, 15, 18, …, 87, 90 (30 cards)
– Multiples of 4: 4, 8, 12, 16, 20, 24, …, 84, 88 (22 cards)
– Multiples of 5: 5, 10, 15, 20, 25, 30, …, 85, 90 (18 cards)
2. Now, let’s count how many times each card is flipped:
– Flipped once: (3 or 5 but not 4), (3 and 4 but not 5), (4 and 5 but not 3)
– Flipped twice: (3 and 5 but not 4), (3 and 4 and 5)
– Flipped three times: none
3. Let’s count each category:
– Flipped once:
* Multiples of 3 only: 20 cards
* Multiples of 4 only: 11 cards
* Multiples of 5 only: 8 cards
– Flipped twice:
* Multiples of 3 and 5: 6 cards
* Multiples of 3, 4, and 5: 3 cards
4. Total cards flipped an odd number of times (which end up orange side up):
20 + 11 + 8 + 3 = 42 cards
Therefore, after Alice has completed all the steps, 42 cards have their orange side facing up.
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