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

a1call · 2016-03-20, 21:01

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

10:2:2:14:15:3:9:0:12:0:0:11:5:11:8₁₉ mod 18₁₉

Hint:Should be able to do it without calculators, pen or pencil.

Batalov · 2016-03-20, 21:30

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

Now, what is the same number mod 20?

a1call · 2016-03-20, 21:36

Quote:

Originally Posted by Batalov

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

Now, what is the same number mod 20?

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.

science_man_88 · 2016-03-20, 21:42

Quote:

Originally Posted by a1call

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.

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 ?

a1call · 2016-03-20, 22:01

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.

xilman · 2016-03-21, 08:20

Quote:

Originally Posted by a1call

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.

Why do you think base-60 was so popular a while back?

davar55 · 2016-04-25, 07:50

Quote:

Originally Posted by a1call

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.

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.

a1call · 2016-04-30, 07:38

Quote:

Originally Posted by davar55

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.

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

a1call · 2016-04-30, 15:42

* 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:

1:0:0₁₉=361

Equal decimal digits (3)

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

Thank you in advance.

retina · 2016-04-30, 16:00

Quote:

Originally Posted by a1call

* 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)

Yes. Easy.

10:10:10₁₁ = 1330₁₀

science_man_88 · 2016-04-30, 16:05

Quote:

Originally Posted by a1call

* 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:

1:0:0₁₉=361

Equal decimal digits (3)

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

Thank you in advance.

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".

2016-03-20, 21:01	#1
a1call "Rashid Naimi" Oct 2015 Remote to Here/There 3·5·137 Posts	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? 10:2:2:14:15:3:9:0:12:0:0:11:5:11:8₁₉ mod 18₁₉ Hint:Should be able to do it without calculators, pen or pencil.

2016-03-20, 21:30	#2
Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 3⁶×13 Posts	Same as mod 9 for decimals. (Sum of digits, then repeat again if needed.) Now, what is the same number mod 20?

2016-03-20, 22:01	#5
a1call "Rashid Naimi" Oct 2015 Remote to Here/There 3×5×137 Posts	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.

2016-04-30, 15:42	#9
a1call "Rashid Naimi" Oct 2015 Remote to Here/There 3×5×137 Posts	* 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: 1:0:0₁₉=361 Equal decimal digits (3) Could the left side have more decimal digits for other numbers/higher-bases? Thank you in advance.