Question

A small number of cards have been lost from a complete pack if I deal the rest of the cards amongst four people three cards remain. If instead I deal them amongst three people two cards remain. And if I deal them amongst five people, two cards remain. How many cards are there?

Answer 1

The number of cards in the deck now is 47.

Step-by-step explanation:

Let us assume there are a total of m cards and  m< 52.

If cards are divided by 4 the number of cards are left undivided = 3

⇒ 4 K +  3  = m   ( K : Card given to each player)

So,  minimum value of m can be 7

Possible values of m = {7,11,15,....,35,39,43, 47,51} ... (1)

As they all leave 3 cards when divided into 4 people equally.

If cards are divided by 3 the number of cards are left undivided = 2

⇒ 3 S +  2  = m   ( S : Card given to each player)  

So,  minimum value of m can be 5.

Possible values of m = {5,8,11,....,38,41,44, 47,50} ... (2)

As they all leave 2 cards when divided into 3 people equally.

If cards are divided by 5 the number of cards are left undivided = 2

⇒ 5 L +  2  = m   ( L : Card given to each player)  

So,  minimum value of m can be 7.

Possible values of m = {7,12,17,....,37,42, 47} ... (3)

As they all leave 2 cards when divided into 5 people equally.

Now, as we can see in all the three sets, the value of m = 47 is in common in all sets.

Hence, the number of cards in the deck now is 47.