10:2:2:14:15:3:9:0:12:0:0:11:5:11:8_19 mod 18
 

What is the fastest way of calculating the following base 19 formula?

10:2:2:14:15:3:9:0:12:0:0:11:5:11:8[SUB]19[/SUB] mod 18[SUB]19[/SUB]


[SPOILER]Hint:Should be able to do it without calculators, pen or pencil.[/SPOILER]

Same as mod 9 for decimals. [SPOILER](Sum of digits, then repeat again if needed.)[/SPOILER]

Now, what is the same number mod 20?

[QUOTE=Batalov;429693]Same as mod 9 for decimals. [SPOILER](Sum of digits, then repeat again if needed.)[/SPOILER]

Now, what is the same number mod 20?[/QUOTE]
Correct,
Assuming the question is addressed to me. I can get the result but same as 11 for decimals, I don't know a short cut.

[QUOTE=a1call;429694]Correct,
Assuming the question is addressed to me. I can get the result but same as 11 for decimals, I don't know a short cut.[/QUOTE]
hint:if what came to my mind is correct it's not all that hard to work one out. edit: I don't even know why I thought what I came up to was wrong . three more hints to think about:

what is b mod b+1 if not b ?
what is that value to an even power ?
what is that value to an odd power ?

Guess I will leave that to others to solve.
But I will comment:
Write numbers in base 31, and easily calculate reminders to 2, 3, 5, 6, 10, 15 and 30.:smile:

[QUOTE=a1call;429697]Guess I will leave that to others to solve.
But I will comment:
Write numbers in base 31, and easily calculate reminders to 2, 3, 5, 6, 10, 15 and 30.:smile:[/QUOTE]Why do you think base-60 was so popular a while back?

[QUOTE=a1call;429694]Correct,
Assuming the question is addressed to me. I can get the result but same as 11 for decimals, I don't know a short cut.[/QUOTE]

The shortcut for divisibility by 20 in base 19 is similar to that of
divisibility by 11 in base 10: add the digits, but with alternating
signs for adjacent place values, then see whether that result is
divisible by 20. For example, 19^2 + 3*19 + 2 is divisible by 20
because 1 - 3 + 2 = 0 is divisible by 20.

[QUOTE=davar55;432487]The shortcut for divisibility by 20 in base 19 is similar to that of
divisibility by 11 in base 10: add the digits, but with alternating
signs for adjacent place values, then see whether that result is
divisible by 20. For example, 19^2 + 3*19 + 2 is divisible by 20
because 1 - 3 + 2 = 0 is divisible by 20.[/QUOTE]

Didn't know that or the former.
Thank you.

* Could writing an arbitrary number in a base higher than 10 in the format dd:dd:.. ,result in a representation with more decimal digits than if written in base 10? (the separators ":" not to be counted)

Example:

1:0:0[SUB]19[/SUB]=361

Equal decimal digits (3)

Could the left side have more decimal digits for other numbers/higher-bases?


Thank you in advance.

[QUOTE=a1call;432839]* Could writing an arbitrary number in a base higher than 10 in the format dd:dd:.. ,result in a representation with more decimal digits than if written in base 10? (the separators ":" not to be counted)[/QUOTE]Yes. Easy.

10:10:10[sub]11[/sub] = 1330[sub]10[/sub]

[QUOTE=a1call;432839]* Could writing an arbitrary number in a base higher than 10 in the format dd:dd:.. ,result in a representation with more decimal digits than if written in base 10? (the separators ":" not to be counted)

Example:

1:0:0[SUB]19[/SUB]=361

Equal decimal digits (3)

Could the left side have more decimal digits for other numbers/higher-bases?


Thank you in advance.[/QUOTE]



akin to asking can a low decimal number ever be represented as a sum with a higher power of another base. or if logint in pari allows itself to be higher for higher bases given the same decimal number:

the real problem then is you can represent more numbers under any given power of a higher number:


19^n = 1.9^n*10^n so if n>3.58 it exceeds the next power of 10 for example so if the first base 19 digit is more than 3 it's actually able to represent a higher number in less "digits".