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 accomodate more than 150 eggs.
So, how many eggs were there?
Answers
Answer:119
Step-by-step explanation:
Let the number
of eggs be a. Then working backwards, we see that a is less than or equal to 150:
If counted in sevens, nothing will remain, which translates to a = 7p + 0, for
some natural number p. If counted in sixes, a = 6 q +5, for some natural number q.
If counted in fives, four will remain. It translates to a = 5w + 4, for some natural
number w.
If counted in fours, three will remain. It translates to a = 4s + 3, for some natural
number s.
If counted in threes, two will remain. It translates to a = 3t + 2, for some natural
number t.
If counted in pairs, one will remain. It translates to a = 2u + 1, for some natural
number u.
That is, in each case, we have a and a positive integer b (in our example,
b takes values 7, 6, 5, 4, 3 and 2, respectively) which divides a and leaves a remainder
r (in our case, r is 0, 5, 4, 3, 2 and 1, respectively), that is smaller than b
The moment we write down such equations we are using Euclid’s division lemma,
which is given in Theorem
Getting back to our puzzle, do you have any idea how you will solve it? Yes! You
must look for the multiples of 7 which satisfy all the conditions. By trial and error
(using the concept of LCM), you will find he had 119 eggs.