MathDB

Adam, Bendeguz, Cathy, and Dennis all see a positive integer

n

. Adam says, "

n

leaves a remainder of

2

when divided by

3

." Bendeguz says, "For some

k

n

is the sum of the first

k

positive integers." Cathy says, "Let

s

be the largest perfect square that is less than

2n

. Then

2n - s = 20

." Dennis says, "For some

m

, if I have

m

marbles, there are

n

ways to choose two of them." If exactly one of them is lying, what is

n

Problem Statement