Odd Or Even?

Number Theory Level 2

Is this number odd or even ?

$\large (122333444455555666666555554444333221)_7$

Clarification: The 7 subscript indicates that we are working in base 7.

No calculators please! :)

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Odd Even

2 solutions

Geoff Pilling
Apr 28, 2016

Relevant wiki: Number Base - Converting to Different Bases

Unlike base 10 numbers, for base 7, you need to count up the number of odd digits in the number. This is because each digit, $a_n$ , represents $a_n*7^n$ , and $7^n$ is odd for all $n$ . So, if there are an odd number of odd digits, the number is odd, otherwise, the number is even. In this number, there are 18 odd digits, so the number is $\boxed{even}$ .

I solved it as well using a similar method! I knew since an odd x odd = odd, and even + odd = odd, that given the first number is a 1, that as the digits increase the parity of the number would alternate (then I proceeded to very sillily move my mouse to the first digit, then slowly move it down going "odd, even, odd, even, ..." because of it being so early in the morning and I didn't want to just add up the digits (knowing that every digit n here repeated n times), ahahaha!

Andrew Tawfeek - 5 years, 1 month ago

Prove that all you need to do is count the number of odd digits.

Alex G - 5 years, 1 month ago

Good point. I have added to my explanation above.

Geoff Pilling - 5 years, 1 month ago

The number is symmetric so the answer is trivially true for every odd base

Esp Ter - 4 years, 7 months ago

Esp Ter
Oct 25, 2016

Since the number has even digits and is palindromic, after expanding, sum of value of the first and last digit is even, second and second last number is even and so on due to the same parity.

Good observation!

If by symmetric you mean palindromic, though, this doesn't work for all symmetric numbers. Only for symmetric numbers with an even number of digits, or an odd number of digits with an even number in the middle. For example $252_7$ symmetric but odd.

Geoff Pilling - 4 years, 7 months ago

Yes. I meant even number of digits. Thanks I'll add that clarification.

Esp Ter - 4 years, 7 months ago

Odd Or Even?

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