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MathematicalPhysicist
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What is the rule of division by 23?
Thanks in advance.
Thanks in advance.
loop quantum gravity said:What is the rule of division by 23?
Thanks in advance.
loop quantum gravity said:Well another question of mine is how do you prove these division rules, do we know for every prime number the rule of division, or not?
loop quantum gravity said:Well another question of mine is how do you prove these division rules, do we know for every prime number the rule of division, or not?
robert Ihnot said:[tex]\sum_{k=0}^{m}a_{k}10^k=3n[/tex]
Since 10 ==1 Mod3, so are all its powers.
loop quantum gravity said:I'm not sure I understand, I need to prove that it's divisible iff [tex]a_0+...+a_m[/tex] is a multiple of 3.
What does the fact that the residue of 10 by 3 is 1 helps me here?
tiny-tim said:Because if a = b = 1 (mod3), then ab = 1 (mod 3) …
so 10n = 1 (mod3),
and ∑ an10n = ∑ an (mod3)
jambaugh said:If you're looking at a last digit rule you need only consider:
If (p,10)=1 then
[tex] n = 10m + r = ap \equiv m \mp kr =bp[/tex]
where k comes from a two digit multiple of p which is one away from a multiple of 10.
[tex]sp = 10k \pm 1[/tex]
[tex] 10bp = 10m \mp 10k r = 10m+ r \mp (10k\pm 1)r = n \mp (10k\pm 1)r = n \mp sbp[/tex]
Thus
[tex] n = 10bp \pm sbp[/tex]
jambaugh said:You could also generate rules from multiples of p more than one away from a multiple of 10 but these would require you multiply the remaining digits of the number n by that number which is harder.
CRGreathouse said:It's clear that numbers n appearing in the factorization of 9, 99, 999, ... have tests of the second sort, since 10^k = 1 (mod n), and so digits can be added in groups of k. So 11 has a divisibility test based on adding digits in pairs, for example.
…
My question: Are there other kinds of tests than those listed here (or equivalent to these)?
CRGreathouse said:[...]
Good point. I'm not sure that those sorts of rules would be practical compared to long division,though.
jambaugh said:Oh! You want practical!
The Rule of Division by 23 is a mathematical concept that states when a number is divided by 23, the remainder will always be either 0, 1, 2, 3, ..., 20, or 22. In other words, the remainder can never be 21.
This rule can be useful in many mathematical applications, such as checking for errors in calculations or determining whether a number is divisible by 23 without actually performing the division. It can also be used in coding and programming to optimize algorithms and processes.
Yes, the Rule of Division by 23 is always true for any integer divided by 23. This rule is a special case of a larger mathematical concept called modular arithmetic.
The Rule of Division by 23 was likely discovered through trial and error or by observing patterns in numbers. It has been around for centuries and was used by ancient civilizations such as the Babylonians and Egyptians in their mathematical calculations.
Yes, there are similar rules for other numbers, such as the Rule of Division by 7 or the Rule of Division by 9. These rules can be discovered through the same methods as the Rule of Division by 23 and can be useful in various mathematical and computational contexts.