GFG App
Open App
Browser
Continue

# Puzzle 45 | Gem in Pockets

Vijay Dinanath Chauhan aka â€śVDCâ€ť 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 â€śnumber of packetsâ€ť, that was equal to the â€śnumber of pocketsâ€ť. In each packet there was a â€śnumber of gemsâ€ť; and the â€śnumber of gemsâ€ť was equal to the â€śnumber of packetsâ€ť.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.Â  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 â€śno. of packetsâ€ť, and the â€śnumber of packetsâ€ť was equal to the â€śno of pocketsâ€ť.

There for the total no. of gems = NxNxN â€“(N) (for the lost packet), i.e. equal to N3-N. N3-N = N(N2-1) = 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, 4â€¦…and so on, the expression yields 6, 24, 60â€¦…Â  where all numbers are divisible by 6.