Question

Consider the following folk puzzle:
A trader was moving along a road selling eggs.An idler who didn't have much to do started to get the trader into a wordy duel. This grew into a fight, he pulled the basket with eggs and dashed it on the floor.The eggs broke.The trader requested the Panchayat to ask the idler to pay for the broken eggs.The Panchayat asked the trader how many eggs were broken.He gave the following response:
If counted in pairs one will remain ;
If counted in threes, two will remain;
If counted in fours, three will remain;
If counted in fives, four will remain;
If counted in sixes, five will remain;
If counted in sevens, nothing will remain;
My basket cannot accommodate more than $150$ eggs.

So, how many eggs were there?

Answer 1

Let eggs=x
When divided by 2,3,4,5 and 6, the remainder is always one less than the divisor
Thus, x is one less than LCM(2,3,4,5,6) OR one less than a multiple of the same.

Thus, x is one less than LCM(2,3,4,5,6) OR one less than a multiple of the same.
LCM=60
now,
x= (60-1) OR (120-1) OR (180-1)
60-1= 59
NOT divisible by 7
120-1=119
Maybe x= 119
180-1= 179
Cannot be over 150.
We can safely conclude that x= 119
The trader was carrying 119 eggs.