2013 Canadian Computing Competition, Stage 2
Day 1, Problem 1: All your base are belong to palindromes
Most of the time, humans have 10 fingers. This fact is the main reason that our numbering system is base-10: the number 257 really means 2 × 102 + 5 × 101 + 7 × 100. Notice that each digit in base-10 is in the range from 0...9.
Of course, there are other bases we can use: binary (base-2), octal (base-8) and hexadecimal (base-16) are common bases that really cool people use when trying to impress others. In base-b, the digits are in the range from 0...b-1, with each digit (when read from right to left) being the multiplier of next larger power of b.
So, for example 9 (in base-10) is:
- 9 in base-16
- 11 in base-8 (1 × 81 + 1 × 80 = 9)
- 1001 in base-2 (1 × 23 + 0 × 22 + 0 × 21 + 1 × 20 = 9)
Noticing the above, you can see that 9 is a palindrome in these 3 different
bases. A palindrome is a sequence which is the same even if it is
written in reverse order: English words such as
racecar are palindromes, and numbers like 9, 11, 1001 are also
Given a particular number X (in base-10), what bases b (2 ≤ b < X) cause the representation of X in base-b to be a palindrome?.
There will be one line, containing the integer X (2 ≤ X ≤ 109).
For test cases worth 80% of the points, you may assume X ≤ 10000.
The output should consist of a sequence of increasing integers, each on its own line, indicating which bases have the property that X written in that base is a palindrome. Note that we will only concern ourselves with bases which are less than X, and that the first possible valid base is 2.
The number 9 was shown to be a palindrome in base-2 and base-8 in the problem description. The other bases do not lead to palindromes: for example, in base-3, 9 is expressed as 100, and in base-5, 9 is expressed as 14.
Point Value: 10 (partial)
Time Limit: 2.00s
Memory Limit: 16M
Added: May 19, 2013
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