a small number of cards has been lost from a complete pack. If I deal among four people, three cards remain. If I deal among three people, two remain and if I deal among five people, two cards remain. How many cards are there?
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A small number of cards has been lost from a complete pack. If I deal with four people, three cards remain. If I deal with three people, two remain and if I deal with five people, two cards remain. How many cards are there?
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3 Answers
Alan Martin
Answered April 13, 2019
Well, let’s start with the five, as this is the most useful. This means that the last digit can only be a 2 or a 7. With 3 cards remaining when there’s 4 people, that means that the last digit has to be odd, so, it has to be a 7 in the last digit.
Now, we’re looking for a number, which has a last digit of 7, which has a remainer of 3 when divided by 4, and a remainder of 2 when divided by 3.
So, with the remainder of 2 when divided by 3, we are looking at 17 (which has a remainder of 3) + 30x (so, this may be 17, 47, 77, 107, 137, 167, etc.)
Now, our 4 remainder 3 means that the formula is 7 + 40y (so, 7, 47, 87, 127, 167)
So, 47 fits on both of them. So, that’s likely the answer.
Technically, if you expand the definition of “pack” to undefined in the number of cards (instead of being somewhere around 52), we get an overall formula of:
47 + 120x
So, 47, 167, 287, 407