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This problem trains for: AMC-8, AMC-10, Math Kangaroo 7-8, Math Kangaroo 9-10.
Given a positive even integer N, how many of the statements below are TRUE?
There is a base in which the number is written as 10.
There is a base in which the number is a palindrome.
There is a base in which the number is written as 100.
There is a base in which the number is odd.
The number N is written as 10 in base N.
In base N-1 the number N is written as 11, a palindrome.
For the number N to be written as 100 in some base, it has to be a perfect square. Since not all even numbers are perfect squares, the statement is not always true.
If the number is divisible by 2, it will continue to be divisible by 2 in any base. The parity of a number is related to the nature of the quantity that the number represents, not to the writing of the number in some representation. The number cannot become odd as a result of writing it in a different base.