Question

A trader was moving along a road selling eggs. An idler who didn't have much work 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 there are x eggs in the basket.
Then, x = 7 × q 1 + 0 [As number is completely
divisible by 7]
x = 6 × q 2 + 5
x = 5 × q 3 + 4
x = 4 × q 4 + 3
x = 3 × q 5 + 2
x = 2 × q 6 + 1
Now, see the number is divisible by 7 and also
the number of eggs are less than 150.
Consider the number less than 150 and divisible
by 7:
147, 140, 133, 126, 119, 112, 105, 98,.....14, 7.
From these numbers, we see that 119 in divisible
by 7.
Also 119 leaves remainder 5 when divided.
Also, the remainder by 6 is 4 when it is divided
by 5.
It leaves remainder 3 when divided by 4.
It leaves the remainder 2 when divided by 3 and
leaves the remainder 1 when divided by 2.
Hence, the number of eggs in the basket were
119