# How Many Eggs ?

A farmer is taking her eggs to the market in a cart, but she hits a pothole, which knocks over all the containers of eggs. Though she is unhurt, every egg is broken.

So she goes to her insurance agent, who asks her how many eggs she had. She says she doesn’t know, but she remembers something from various ways she tried packing the eggs.

When she put the eggs in groups of two, three, four, five, and six there was one egg left over, but when she put them in groups of seven they ended up in complete groups with no eggs left over.

**What can the farmer and agent figure out from this information about the number of eggs she had ? **

**Is there more than one answer ?**

PS:- Though I know the answer but still I want the answer in the simplest understandable manner possible. Thanks for all answers in advance.

The answer is __301__

__EXPLANATION-__

__Condition 1 – __**When she put the eggs in groups of two, three, four, five, and six there was one egg left over**Since there was one egg left when packed in 2,3,4,5,6, we need to find the LCM (lowest common multiple) of 2,3,4,5,6 and add one to get the

*least*possible answer, THat is 60+1=61.

Moreover, we can easily figure that

**60x+1**will always satisfy the first 5 conditions, where

**x**is a positive integer.

__Condition 2- __**when she put them in groups of seven they ended up in complete groups with no eggs left over**

To satisfy this condition, we have to find the *least* value of **X** such that **60x+1** is divisible by **7**.

That is,

(60x+1)%7 = 0 Where % is mod

When,

x=1 —–> 61 % 7 = 5

x=2 —–>121 % 7 = 2

x=3 —–>181 % 7 = 6

x=4 —–>241 % 7 = 3

__x=5 —–>301 % 7 = 0____ —- SATISFIED__

**Hence, 301 eggs is the Minimum Number of eggs.**

** NOTE-**While there are many possible answers satisfies this equation (60x+1)%7 = 0

__Answers__**–**301, 721, 1141, 1561, 1981 …………

Nice Explanation @ SherlockHolmes. But think over no. 21.

2 * 3 + 3 * 1 + 5 * 1 + 6 * 1 = 20, therefore one egg will be left over.

| |

| no. of such group

group of 2

301, 721, etc.

My explanation:

the number is a multiple of LCM of (2,3,4,5,6) + 1 = 60*x + 1

And this number should be divisible by 7.

It’s easy to see the above number will always end up with the number 1. There’s only one multiple of 7 which ends in 1, i.e., 3 (7*3 = 21).

So we only need to check all prime numbers that end in 3. For example, 7 * 13, 7 * 23, 7 * 43, etc. because only these multiples of 7 will result in a number that ends in 1 and also the multiple has to be prime or else it’ll be divisible by either 2, 3 or 5 negating the above condition of leaving one egg.

so we can check all such numbers multiplied by 7 and subtract 1 and check if the number is also a multiple of 60::

7 * 13 = 91 – 1 = 90 (not a multiple of 60)

7 * 23 = 161 – 1 = 160 (not a multiple of 60)

(33 is not a prime number so we don’t need to check that)

**7 * 43** = **301** – 1 = 300 (multiple of 6) (Answer)

and we can find many more numbers like this

7 * 53

7 * 73

7 * 83

**7 * 103 = 721**

### Your Answer

## More puzzles to try-

### What is the logic behind these ?

3 + 3 = 3 5 + 4 = 4 1 + 0 = 3 2 + 3 = 4 ...Read More »### Five greedy pirates and gold coin distribution Puzzle

Five puzzleFry ship’s pirates have obtained 100 gold coins and have to divide up the loot. The pirates are all ...Read More »### Defective stack of coins puzzle

There are 10 stacks of 10 coins each. Each coin weights 10 gms. However, one stack of coins is defective ...Read More »### Dr. Willam and surgical gloves puzzle

Dr.Willam wants to operate for three different persons who were wounded. But he had only two surgical gloves. There is ...Read More »### Which clock works best?

Which clock works best? The one that loses a minute a day or the one that doesn’t work at all?Read More »### To which one will the needle touch 1 OR 2

In the gear arrangement, To which one will the needle touch, 1 OR 2Read More »### How did he know?

A man leaves his house in the morning to go to office and kisses his wife. In the evening on ...Read More »### Which letter replaces the question mark

Which letter replaces the question markRead More »### Crossing the Bridge Puzzle (Bridge and torch problem)

A bridge will collapse in 17 minutes. 4 people want to cross it before it will collapse. It is a ...Read More »### Probability of having boy

In a country where everyone wants a boy, each family continues having babies till they have a boy. After some ...Read More »### Magical flowers!!

A devotee goes to three temples, temple1, temple2 and temple3 one after the other. In front of each temple, there ...Read More »### How many times a day do all three hands of analog watch overlaps?

There is a perfectly accurate analog watch with hour, minute, and second hands. How many times a day do all three ...Read More »### Tuesday, Thursday what are other two days staring with T?

Four days are there which start with the letter ‘T‘. I can remember only two of them as “Tuesday , Thursday”. ...Read More »### How could only 3 apples left

Two fathers took their sons to a fruit stall. Each man and son bought an apple, But when they returned ...Read More »### Minimum number of persons needed to cross a Desert

In the middle of the confounded desert, there is the lost city of “Ash”. To reach it, you will have ...Read More »### Lateral thinking sequence Puzzle

Solve this logic sequence puzzle by the correct digit- 8080 = 6 1357 = 0 2022 = 1 1999 = ...Read More »### Pizza Cost Math Brain Teaser

Jasmine, Thibault, and Noah were having a night out and decided to order a pizza for $10. It turned out ...Read More »### Which room is safest puzzle

A murderer is condemned to death. He has to choose between three rooms. The first is full of raging fires, ...Read More »### Overthrow your math teacher!

Some numbers are very mysterious. They are all integers. They have more than one digit. If you multiply them with ...Read More »### Richie’s Number System

Richie established a very strange number system. According to her claim for different combination of 0 and 2 you will ...Read More »