20210410, 02:02  #12  
"Rashid Naimi"
Oct 2015
Remote to Here/There
7E5_{16} Posts 
Quote:
Somehow I erred at computing the valuation(OPNumber1,2) as 2 rather than 3. In the integer field if valuation(n1,2) =n' for any positive odd integer then if m==a*b if a'==m' then b'>a' if a'<m' then b'==a' Here my error was that m'==2 rather than 3 and since neither a' nor b' could be less than 2 then I wrongly concluded that one had to be equal to 2. Again, thank you very much for the correction. Much appreciated. Last fiddled with by a1call on 20210410 at 02:15 

20210410, 02:57  #13 
"Nigel"
Apr 2021
2×7 Posts 

20210410, 12:47  #14  
Feb 2017
Nowhere
2×3×757 Posts 
Quote:
And by golly, zero is even! 

20210411, 10:51  #15 
"Nigel"
Apr 2021
1110_{2} Posts 
OP=??
Why did you all start calling the number OPnumber?

20210411, 10:54  #16 
Jan 2021
California
2^{2}×17 Posts 

20210411, 15:59  #17  
Feb 2017
Nowhere
2·3·757 Posts 
Quote:
Exercise: To compute the remainder of n (mod 2^k) or (mod 5^k), it suffices to compute the remainder of the last k decimal digits of n (mod 2^k) or (mod 5^k). 

20210411, 16:43  #18 
"Rashid Naimi"
Oct 2015
Remote to Here/There
43×47 Posts 
I'm afraid you are giving me too much credit for being able to do arithmetic in my head. I actually used Pari and specifically remember getting 2 as the valuation. I can only assume that it must have been a typo somewhere. But thanks for the calculation trick.
ETA Being a CAD designer for most of your life, pretty much strips you of being able to do any arithmetic in your head. It's sort of a "if you don't use, you lose it" in the extreme. Accordingly with the advent voicerecognition and speechtotext software, I expect the future generations not being able to correctly spell a word if their lives depended on it. That would put me as much at home as it would kriesel out. Last fiddled with by a1call on 20210411 at 17:01 
20210411, 17:24  #19 
Romulan Interpreter
Jun 2011
Thailand
9445_{10} Posts 
Yo don't need to do advanced math in your head, what he is referring to is the schoolgrade rules of divisibility, like "a number is divisible by 3 if the sum of digits is divisible by 3", "a number is divisible by 4 if the last two digits are divisible by 4", "a number is divisible by 5 if the last digit is 0 or 5", etc. For 4, any number can be written 100*x+y, where y<100, and 100*x is always divisible by 4, therefore for the number to be divisible by 4, is enough that y is divisible by 4. Ex: 1234 is not divisible by 4, because 34 is not divisible by 4, this number is 2 (mod 4). This works because you can write 1234 as 1200+34, and 1200 is always divisible by 4 (as is multiple of 100, so it is 4 times multiple of 25), so it is enough to examine the last 2 digits. You can apply that for 8, etc.
Edit: For 8 there is even a simpler rule, where you look first if the third digit from the right (the "hundreds", i.e. the 2 in 1234) is odd or even, and according with it, you examine the last two. Can you say why that works, and how do we know at a simple glance that 1234 is 2 (mod 8)? Last fiddled with by LaurV on 20210411 at 17:26 
20210411, 19:18  #20 
Bamboozled!
"πΊππ·π·π"
May 2003
Down not across
2×7^{2}×109 Posts 
To check for divisibility by 7, 11 and 13 simultaneously use the ruleofeleven three digits at a time and examine the result.
I quite often use this to check primality of small numbers (<30030) without needing to use a calculator. If it passes this check, and the simple ones for 2, 3 and 5, mental trial division by 17, 19, 23 and 29 is straightforward and reaches 6469693230. Checking divisibility by 37 is easy enough by casting out the 999s. An additional mental division by 31 suffices to 7420738134810 (which is somewhat over 7 billion). Long before that I tend to resort to computer assistance. 
20210412, 10:30  #21  
Bamboozled!
"πΊππ·π·π"
May 2003
Down not across
24672_{8} Posts 
Quote:
37# is certainly >7.4 billion, and so on, but the procedure outlined work only to 41^21 = 1680 as a primality test. It filters out 85% of composites in the limit. Still, I find it useful. 

20210421, 06:44  #22 
"Nigel"
Apr 2021
1110_{2} Posts 
Another 129 digit number
Code:
483100240063213823403065506870079481290504898733191281212060160586898805944701565058228233588987690426223309716587384614909677337 Last fiddled with by retina on 20210421 at 06:48 Reason: More yucky horizontal scrolling, changed to a code box. You can fix this yourself 
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