# Puzzle 45 | Gem in Pockets

Vijay Dinanath Chauhan aka “VDC” was travelling from Bangalore to New Delhi along with his friends. VDC was wearing a Cargo Jeans with finite number of small pockets. And in his pockets, there were packets of gems, not the real ones,actually the Cadbury gems to eat when he reaches Delhi.In each pocket there was “number of packets”, that was equal to the “number of pockets”. In each packet there was a “number of gems”; and the “number of gems” was equal to the “number of packets”.Unfortunately, when he reaches Delhi and gets down from the train, due to huge crowd he lost one of his Cadbury gems packet.

Shocked by the incident, as he loves the gems most, he decided not to eat the gems that day, and donate all of them to the beggars sitting nearby. But when he went to them, they all started rushing towards him to get the gems. He being a good human wanted to distribute those gems equally. Find the number of beggars so that he can distribute the gems equally?

*Note: No. of beggars>1, No. of pockets>1*

**Solution:** 6,VDC was silly, he could have eaten the gems any other day! Hahaha, Let the number of pockets be N. The number of gems in each packet is equal to the “no. of packets”, and the “number of packets” was equal to the “no of pockets”.

There for the total no. of gems = NxNxN –(N) (for the lost packet), i.e. equal to N^{3}-N. N^{3}-N = N(N^{2}-1) = N(N+1) (N-1). This last expression is divisible by 6 in all cases, since a number is divisible by 6 when it is both divisible by 3 and even. And substituting 2, 3, 4……and so on, the expression yields 6, 24, 60…… where all numbers are divisible by 6.

This puzzle is contributed by** Praveer Satyam. **Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above