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Suppose all of them plucked different number of apples.A boys can plucks at least 0 apples and the number of apples plucks by each student is distinct.

So,Total number of plucked apples should be less then 100.

but 0+1+2.....+13+14 =105>100  a contradiction.

Thus there exist two boys who plucked the same number of apples. 

If Thus there exist two boys who plucked the same number of apples.It means each boy of remaining 13 students plucked different number of apples..

Number of apples Plucked by 13 student = 0+1+2+3...+12= 78 Apples

Number of apple plucked by remaining 2 student = 100 -78 =22

Both student plucked same number of apples.

So, Number of apples plucked by one of them = $\frac{22}{2}=11$ 

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