The Egg Problem
Once upon a time there was a man who sold eggs for his livelyhood. One day he was going to sell those eggs and suddenly collide to a drunkardand all eggs were smashed.So he complaint to head of village and he asked him how many eggs were there.In response man replied,"I never count them but i know that when eggs were grouped in pairs 1 egg left behind. Simlarly for triplets ,2 were left behind and for four ,3 were left behind.For five,4 were left behind and for six ,5 were left behind. But when grouped in seven ,no egg is left behind. Also my egg basket cannot hold more than 150 eggs." So can you help him finding the number of eggs?
The answer is 119.
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Egg quantity
= n x { LCM (2, 3, 4 & 5) } - 1
= 60n - 1
= 7k
The number must have a last digit of 9 (from 60n - 1) so k must have a last digit of 7.
60n - 1 = 7(10m + 7)
60n = 70m + 50
for 0 < n < (150 + 1) / 60 & 0 ≤ m < (150 - 49) / 70
==> 0 < n < 2.5 & 0 ≤ m < 1.4
6n = 7m + 5
Since m must be of odd parity to even out the summand of odd 5, m must be 1 as the only possible answer in the range above.
n = (7 x 1 + 5) / 6 = 2
Answer
= 60 x 2 - 1
= 119