11th May 2023:

‘n’ mangoes are to be distributed in such a way that following conditions are satisfied:
(i) When mangoes are distributed among 2 persons, then one mango is left out
(ii) When mangoes are distributed among 3 persons, then also one mango is left out
(iii) When mangoes are distributed among 4 persons, then also one mango is left out
(iv) When mangoes are distributed among 5 persons, then also one mango is left out
(v) When mangoes are distributed among 6 persons, then also one mango is left out
(vi) When mangoes are distributed among 7 persons, then each one gets equal number of mangoes.
The least number of mangoes required satisfying above conditions is ___.

Solution:
As per the question, distribution of ‘’ mangoes is done keeping two conditions in mind –
1. Mangoes are to be distributed equally amongst 2, 3, 4, 5, 6 people leaving 1 mango as extra.
2. Equal distribution of mangoes amongst 7 people with no mangoes being left out.

The required number leaves the remainder 1 when it is divided by 2, 3, 4, 5, 6 and 0 remainder when divided by 7.
Therefore, the required number is the least common multiple of 2, 3, 4, 5 & 6 leaving 1 as remainder.
i.e., number is of the form , which must be divisible by 7 leaving 0 as remainder.

Substituting the values of  sequentially, we get
 is not divisible by 7
 is not divisible by 7
 is not divisible by 7
 is not divisible by 7
 is divisible by 7

 The least number of mangoes required to distribute is 301 satisfying all the conditions.