63. Equal Candy Surprise
A father brings home packets of candies for his six daughters.
- 1 packet for the 1st daughter
- 2 packets for the 2nd daughter
- 3 packets for the 3rd daughter
- 4 packets for the 4th daughter
- 5 packets for the 5th daughter
- 6 packets for the 6th daughter
Each daughter discovers that, when she adds up the candies in all of her own packets, she has exactly the same total number of candies as every other sister.
The candies in any one packet are all whole candies, and every packet given to the same daughter contains the same number of candies (though different daughters may have packets of different sizes).
What is the smallest possible number of candies in each packet for every daughter?
Added 7 October 2009
Hint: Think about the least common multiple of the packet counts 1, 2, 3, 4, 5 and 6.
Solution:
Let T be the common total number of candies each daughter receives. For the totals to be whole numbers of candies per packet we need T to be divisible by each daughter’s packet count 1, 2, 3, 4, 5 and 6. The least common multiple of those numbers is 60, so the smallest possible total is T = 60 candies.
Then the candies in each packet are:
- 1st daughter: 60 candies in her single packet (60 ÷ 1)
- 2nd daughter: 30 candies in each of her two packets (60 ÷ 2)
- 3rd daughter: 20 candies in each of her three packets (60 ÷ 3)
- 4th daughter: 15 candies in each of her four packets (60 ÷ 4)
- 5th daughter: 12 candies in each of her five packets (60 ÷ 5)
- 6th daughter: 10 candies in each of her six packets (60 ÷ 6)
Thus the smallest feasible distribution is 60 candies each, with packet sizes of 60, 30, 20, 15, 12 and 10 candies respectively.
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