Even and Odd Numbers in Base 5
Date: 02/02/2002 at 15:35:59 From: Sheri Griffiths Subject: Even and odd numbers in base 5 How can you tell if a number in base 5 is even or odd? I know that all even numbers in base 10 are divisible by 2, but I don't know if it translates over exactly to base 5. I want to know if there are conversions from base 10 to base 5 to be able to figure out this answer.
Date: 02/02/2002 at 17:32:32 From: Doctor Rick Subject: Re: Even and odd numbers in base 5 Hi, Sheri. I think of "even" as meaning "divisible by 2," and this is independent of the base in which we represent a number. But perhaps you are thinking of "even" as meaning "having a units digit that is divisible by 2." It's true that not all numbers whose units digit in base 5 is even are themselves even (that is, divisible by 2). The even numbers are Base 10: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, ... Base 5: 0, 2, 4, 11, 13, 20, 22, 24, 31, 33, ... You will notice that in every case in this list, the SUM of the digits of the base-5 numeral is even. We can't say yet that this rule holds true, because we haven't looked at any 3-digit numerals; and even then we won't be sure until we actually prove the rule. Here is how we can prove our rule. The numeral abcd in base 5 represents the number 125a + 25b + 5c + d The powers of 5 are all odd, so we can write the number like this: (2*62+1)a + (2*12+1)b + (2*2+1)c + d 2(62a+12b+2c+d) + (a+b+c+d) The first term must be even. If the second term is even, then abcd will be even; and if the second term is odd, then abcd will be odd. Thus our hypothesized rule works: a numeral in base 5 is even or odd according as the sum of its digits is even or odd. Since *all* powers of 5 are odd, the proof can be extended to any number of digits. In fact, the same rule holds for all odd bases. Do you see why? We have a briefer answer to this question, with a nice addition that makes the rule easier to use, in our Dr. Math Archives: Even or Odd in Base 5? http://mathforum.org/dr.math/problems/susan.9.23.99.html - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
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