Problem #PRU-97855

Problems Number Theory Divisibility Division with remainders. Arithmetic of remainders Division with remainder Methods Pigeonhole principle Pigeonhole principle (other)

13–16 4.0

Problem

A group of numbers $A_{1}, A_{2}, \dots, A_{100}$ is created by somehow re-arranging the numbers $1, 2, \dots, 100$ .

100 numbers are created as follows: $B_{1} = A_{1}, B_{2} = A_{1} + A_{2}, B_{3} = A_{1} + A_{2} + A_{3}, \dots, B_{100} = A_{1} + A_{2} + A_{3} \dots + A_{100} .$

Prove that there will always be at least 11 different remainders when dividing the numbers $B_{1}, B_{2}, \dots, B_{100}$ by 100.

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