Stolen Mangoes

                 Three naughty boys stole some mangoes from a garden. As it was late in the evening, they decided to divide the fruit equally among them in the morning, and went to sleep.

                At night while the other two were sleeping one boy woke up, tip- toed to the basket of mangos, counted them and is one. From the remainder he took a precise third and went back to sleep.

               After some time a second boy woke up. He counted the mangoes, ate one, took an exact third of the remaining and went back to sleep.

              A little before sun rise the third boy also woke up, ate one, and like the other two boys took a precise third of the remainder in whole mangoes.

              In the morning, all the three boys went together to the basket of mangoes, counted them. Amongst them they found one which was over ripe- almost rotten. They threw it away. From the remainder they made an exact division.

             How many mangoes in all did they steal?

Answer: Let the number of mangoes be X.

The first naughty boy at one, leaving x-1, He left behind two third of

X -1 or 2(x-1) / 3   or     2x -2   / 3

The second naughty boy ate one leaving

2x -2   / 3 - 1

Or    2x – 2-3    /3   or    2x -5   /3

He left behind two – thirds of

 Or 2x -5 / 3 or   2(2x – 5) / 3x3   or

4x – 10 / 9

The third naughty boy ate one, leaving

4x - 19   / 9   -1

Or   4x -10 -19 / 9   or    4x -19/9

He left behind two- thirds of

4x -19 /9   or 2(4x – 19) / 3x 9

Or 4x - 38 / 27

In the morning one mango was eaten leaving

8x - 38 /27 -1 or   8x -38 – 27 / 27 or 8x - 65 / 27

Since 8x – 65 /27

 Mangoes were divided equally

Into three parts, the number must be a multiple of 3.

Let 8x -65 /27     = 3n where n is an integer.

 Then 8x -65 = 81n

8x   = 81n +65

Since x is a whole number, 81n + 65 must be an even number, because no odd number is divisible by 8.

Since 65 are odd 81 must be odd, because the sum of two odd numbers is even.

Since 81 n is odd, n must be odd.

Let n be equal to the odd number 2b + 1

Then 8x = 162b + 81 + 65

Or    8x = 162b +146

 Or    4x   =81b + 73

As before the right side must be even.

Therefore b must be odd.

Let b be equal to the odd number 2c + 1

Then 4x   = 162c +81 +73

Or    4x = 162c + 154

Or   2x = 81c +77

As before the right side must be even.

Therefore c must be odd.

Let c be equal to the odd number 2d + 1

Then 2x   = 162d + 81+77

Or    2x = 162d + 158

Or   2x =81d +79

The least value of X will be obtained when d = 0

 Then x = 79

The next value of x can be obtained when x= 1

Then x=160

The next value of x when d= 3 will be 241.

But by verification 79 will be the correct answer.