Century Basket
Posted on Saturday, June 20th, 2009 at 1:29 pmhow many eggs in the basket?
(From Brahmagupta, 7th century A.D.) A girl was carrying a basket of eggs, and a man driving a horse hit the basket and broke all the eggs. Wishing to pay for the damage, he asked the girl how many eggs there were. The girl said she did not know, but she remembered that when she counted them by twos, there was one left over; when she counted them by threes, there were two left over; when she counted them by fours, there were three left over; when she counted them by fives, there were four left; and when she counted them by sixes, there were five left over. Finally, when she counted them by sevens, there were none left over. `Well,' said the man, `I can tell you how many you had.' What was his answer?
This is just a simple application of the Chinese remainder theorem.
Let N denote the number of eggs. Then:
N ≡ 1 (mod 2)
N ≡ 2 (mod 3)
N ≡ 3 (mod 4)
N ≡ 4 (mod 5)
N ≡ 5 (mod 6)
N ≡ 0 (mod 7)
Which is equivalent to:
N ≡ -1 (mod 2)
N ≡ -1 (mod 3)
N ≡ -1 (mod 4)
N ≡ -1 (mod 5)
N ≡ -1 (mod 6)
N ≡ 0 (mod 7)
From this we can make use of LCM(2,3,4,5,6) = 60 to obtain:
N ≡ -1 (mod 60)
N ≡ 0 (mod 7)
Finally, we simply write:
N = 60k - 1
Reduce modulo 7 to get:
N ≡ 4k - 1 ≡ 0 (mod 7)
4k ≡ 1 (mod 7)
k ≡ 2 (mod 7)
Now the least k could be is 2, which gives:
N = 119
The next largest possibility is k = 9, which would give
N = 539
which seems to be too many for a little girl to carry!
Answer: 119 eggs.
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