7 thieves steal diamonds from a royal treasury and escape the kingdom. As darkness sets they take shelter in a cottage inside a jungle. Two of them get up in middle of the night and split the diamonds between themselves but that left one remainder. So they woke the third thief and split the diamonds between themselves which left one remainder again. They wake up the fourth one and split the diamonds again but find that leaves one remainder. Wake up the fifth thief, split the diamonds between themselves but find one remainder agaian. Wake up sixth, split the diamonds only to find that leaves one remainder again. Ended up waking up the seventh guy and split. That shares equally. How many diamonds did they steal?
Answers
AnSwEr : -
- Number of diamonds is either 301 or 721
GiVeN : -
- Total number of Thieves = 7
- When Two , three, four, five, six of them divide the diamonds respectively then in each case it leaves same remainder 1.
- When seven thieves divide it then it divides completely without leaving any remainder.
To FinD : -
- Total number of Diamonds stolen
SoLuTiOn : -
In the questions it is given that it is equally shared when it divides with 7 .
So,
No of diamond should be multiple of 7 and does not divide by number that is less than 7 .,(i. e. 2,3,4,5,6)
Those numbers which are less than 7 when divides the number gives 1 as a reminder in each case.
Now,
L. C. M of 2,3,4,5,6 = 60
So,
We have some numbers which are common multiples of 2,3,4,5,6:-
=> 60 , 120 , 180 , 240 , 300 , 360 , 420 , 480 , 540 , 600 , 660 , 720,etc
This forms an AP this means every multiple of 60 is divided by 2,3,4,5,6 but not by 7
But only 2 of them satisfy the conditions of questions :-
Those two numbers are 300 and 720.
Now,
In the question it is given that when each number divides this no gives 1 as a remainder in each case so we have to add 1 in this number.
1st number = 300
Add 1 in this number = 300 + 1 = 301
Second Number = 720
Add 1 in this Number = 720 + 1 = 721
Now,
When this number divides by 2, 3 , 4 ,5 , 6, 7 , it gives :-
301 = 2 × 150 + 1
301 = 3 × 100 + 1
301 = 4 × 75 + 1
301 = 5 × 60 + 1
301 = 6 × 50 + 1
Now the above numbers when divides this number gives 1 in each case.
Now,
301 = 7 × 43 + 0
Now it is completely divisible by 7 and not by 2,3,4,5,6.
Now,
Second Number :- 721
721 = 2 × 360 + 1
721 = 3 × 240 + 1
721 = 4 × 180 + 1
721 = 5 × 144 + 1
721 = 6 × 120 + 1
Now in the above cases we get 1 as a remainder in all cases.
Now we have,
721 = 7 × 103 + 0
Hence it is divided by only 7 completely without leaving any remainder.
Therefore,