Five children collect N pieces of Halloween candy and decide to split it evenly

Question

Five children collect N pieces of Halloween candy and decide to split it evenly among them. When they try to divide it they have two pieces of candy left over. One of the children leaves, taking the 26 pieces of candy she collected with her. The remaining four children try to split the N

Explanation / Answer

The first sentence of the question says when N is divide by 5 it leaves a remainder 2. 2nd sentence says when N-26 is divided by 4 it leaves a remainder 1. Also when N-50 is divided by 3 it leaves a remainder 0. 26 mod 4 = 2 => N mod 4 = 3 50 mod 3 = 2 => N mod 3 = 2 (as (N-2-48) mod 3 =0 ) N mod 5 = 2 Thus N should be such a number that N-2 is a multiple of 5, N-3 should be a multiple of 4, N-2 is a multiple of 3. Also, they try to split the candies equally, so when 1st child leaves remaining candies for each child should be greater than 26*4 (since the 1st child cannot have more candies than the remaining children). therefore N > 26 + (26*4) = 105 Since the second child leaves with only 24 candies, The above mentioned condition is more effective. Thus N is a whole number > 105 and satisfies the above 3 mod conditions. Therefore the nearest value for N is 107. Now the next possible for N is when it satisfies the above mod conditions, this occurs at the Least Common Multiple of 5, 4 and 3 i.e. 60. therefore next possible values of N are 107 + 60 n (where n is any natural number)

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