Question on Number of Mangos in a Basket

Topic : Number of Mangos in a Basket

Question : A mango seller describes the number of mangoes in his basket as :
IF the mangoes are grouped in twos, then 1 mango remains
IF the mangoes are grouped in threes, then 2 mangoes remain
IF the mangoes are grouped in fours, then 3 mangoes remain
IF the mangoes are grouped in fives, then 4 mangoes remain
IF the mangoes are grouped in six, then 5 mangoes remain
IF the mangoes are grouped in sevens, then no mangoes left.
Find the number of mangoes in the basket.

Solution :
The least number of mangoes would be 119.
but if the question is to find the number of mangoes then119,539,959,1379,1799,2219,..... all qualify for the answer... as shown below.
Let n denote the number of mangoes in the basket.
then since n leaves remainder 1 when divided by 2, hence
2 divides n+1
since n leaves remainder 2 when divided by 3, hence
3 divides n+1
since n leaves remainder 3 when divided by 4, hence
4 divides n+1
since n leaves remainder 4 when divided by 5, hence
5 divides n+1 since n leaves remainder 5 when divided by 6, hence
6 divides n+1
since 2,3,4,5,6 divide n+1 hence
the LCM of 2,3,4,5,6 = 60 divides n+1
hence n+1 = 60k for some positive integer k . [ we need to find k now]
also since 7 divides n hence
60k leaves remainder 1 when divided by 7, hence
60k=1(mod 7) , which gives
4k=1(mod 7), which gives
k=2(mod 7).
Hence k = 2,9,16,23,30,37......
Hence n+1=60k gives
n = 119,539,959,1379,1799,2219,.....
hence the least number of mangoes would be 119.
but if the question is to find the nuber of mangoes then
119,539,959,1379,1799,2219,..... all qualify for the answer.

Note :
The number of mangoes = LCM (2,3,4,5 and 6) - 1 = 120 - 1 = 119 and the sequence is = 120N - 1 where N is any whole number.