Elevens!
 

:smile:
a) Find the largest possible number containing any 9 of the 10 digits (0 included) which can be divided by 11 and leaving no remainder

b) Find the smallest possible number, with the same conditions, divisible fully with no remainder by 11

Eg: 896743012 (5 omitted). This is divisible by 11 but its neither the largest nor the smallest number.

Mally :coffee:

Do you consider numbers starting with "0" to be properly formed?

[QUOTE=Wacky]Do you consider numbers starting with "0" to be properly formed?[/QUOTE]

Define 'properly formed'.

Elevens!
 

:surprised
Not really Wacky as it would make a difference in the smallest number if we considered this permissible.
Mally :coffee:

[QUOTE=R.D. Silverman]Define 'properly formed'.[/QUOTE]
He's asking whether leading zeros are permissible.

Paul

[QUOTE=xilman]He's asking whether leading zeros are permissible.

Paul[/QUOTE]

I'm sorry. I did not/can not see the original post, so I lost the context.
mfgoode is on my 'ignore' list as I believe that trying to have a
rational discussion with him is pointless.

I'm sorry
 

[QUOTE=R.D. Silverman]I'm sorry. I did not/can not see the original post, so I lost the context.
mfgoode is on my 'ignore' list as I believe that trying to have a
rational discussion with him is pointless.[/QUOTE]
:surprised
Thats okay with me R.D. IT suits me fine. Good riddance to bad rubbish. I dont need you anyway as you seldom solve anything in my opinion. As Dr. Peter would say 'You have reached your level of incompetence' and you intend staying there for quite awhile.
The venom you tend to exude with every post will consume you.
No hard feelings and I wish you luck
Mally :coffee:

[spoiler]
102347586
987652413
[/spoiler]

:cat:

Elevens!
 

:bow:
You are absolutely right.
Can you give us the rules you used to obtain this ?
Mally :coffee:

Simple:
for the maximum you start with 98765 and then you try to obey to the eleven division rule. Same for finding the minimum, assuming that you cannot put 0 at the beginning, the best you can do is start with 10234

:cat:

Division rules
 

[QUOTE=Kees]Simple:
for the maximum you start with 98765 and then you try to obey to the eleven division rule. Same for finding the minimum, assuming that you cannot put 0 at the beginning, the best you can do is start with 10234

:cat:[/QUOTE]
:smile:
Thank you Kees thats great!

I presume you mean the division rule that if the sum of the digits in the even places is the same as the sum of digits in the odd places then the number is divisible by 11 without a remainder.

However little is known of the the other rule that if the difference between the sum of the odd and the even digits is 11 or a multiple of 11 then it can equally be applied and is valid.

Regards,
Mally :coffee:

Obviously, I was applying the second rule

Elevens!
 

QUOTE=mfgoode]:surprised
Thats okay with me R.D. IT suits me fine. Good riddance to bad rubbish. I dont need you anyway as you seldom solve anything in my opinion. As Dr. Peter would say 'You have reached your level of incompetence' and you intend staying there for quite awhile.
The venom you tend to exude with every post will consume you.
No hard feelings and I wish you luck
Mally :coffee:[/QUOTE

:mellow:
I'm sorry for those remarks R.D.
When the super heated steam engine exceeds the designed value (220 lbs / sq. in. in my Dad's days) a safety valve blows to release the excess and reduce the pressure so that the boiler does not explode.
Thats exactly what happened to me and I straight away let off steam.
I retract my vitriolic remarks and kindly accept my apologies.
I know it gets lonely at the top
Just to remind you that we all need and admire you for your above average knowledge which is exceptional and keep the fire burning. Even one candle can expel all the darkness that surrounds us all the time.
Mally :coffee:

Rules
 

[QUOTE=Kees]Obviously, I was applying the second rule[/QUOTE]
:cat: Thank you Kees.
Well this thread wont be complete and exhausted without this last problem though I am sure there may be many more. :blush:
here are the nine digits so arranged that they form four square numbers :
9 , 81 , 324 , 576. (using all 9 digits and no more repetitions)
Now could you put them all together so as to form a single square ?
1) the smallest possible ? and
2) the largest possible ?
Mally :coffee:

With this simple program I wrote in UBASIC:

[CODE] 10 dim C(9)
20 for A=10000 to 31622
30 B=A*A
40 C(1)=B@10
50 I=2
60 C(I)=(B\10^(I-1))@10
70 if C(I)=0 then 150
80 J=1
90 if C(I)=C(J) then 150
100 J=J+1
110 if J<I then 90
120 I=I+1
130 if I<10 then 60
140 print A,B
150 next A[/CODE]
I found the 30 squares formed with the digits 1-9:

[CODE] 11826 139854276
12363 152843769
12543 157326849
14676 215384976
15681 245893761
15963 254817369
18072 326597184
19023 361874529
19377 375468129
19569 382945761
19629 385297641
20316 412739856
22887 523814769
23019 529874361
23178 537219684
23439 549386721
24237 587432169
24276 589324176
24441 597362481
24807 615387249
25059 627953481
25572 653927184
25941 672935481
26409 697435281
26733 714653289
27129 735982641
27273 743816529
29034 842973156
29106 847159236
30384 923187456
[/CODE]

It appears that no combination of the numbers 9 , 81 , 324 , 576 are in these squares.

Elevens!
 

:surprised
Excellent Alpertron!
But what about the smallest square using all the 9 digits? I dont see it around in your post. Thanks!
Mally :coffee:

The smallest is 11826[sup]2[/sup] = 139854276 and the greatest is 30384[sup]2[/sup] = 923187456.

Elevens
 

[QUOTE=alpertron]The smallest is 11826[sup]2[/sup] = 139854276 and the greatest is 30384[sup]2[/sup] = 923187456.[/QUOTE]
:bow:
Thanks once again. How stupid of me! It is sitting right on top of your program. I realised this when I quit mersenneforum. By then it was too late to edit my post as the time limit was up.
You have made an interesting point in the same post #15. If you multiply 9,81,324,576 you will get a square naturally but the digits 2,5,7 are missing and so it does not meet the criterion..
Also I came across how to tell a probable square by inspection no matter how Large it is we must see if the last number is one of the foll:
1,4,9,6,5 and it goes on in the reverse sequence as 5,6,9,4,1 and so on and on. Of course this is elementary but few people use this.
In A run through of your program you have given, we can spot a mistake just by checking if the last number is one of those mentioned. Any other digit will highlight that it cannot be a perfect square.
Thanks Alpertron,
Mally :coffee:
P.S. BTW How far do tables on squares go? a million? or 10 million of them? This will be easy by your programs to tabulate them. I mean somehing like Lehmer's table of primes which can be used as a ready reference for those who need it
As You are onto sequences, perhaps you could add these on as part of your
web site and repertoire. Only a suggestion.

[QUOTE=mfgoode]Also I came across how to tell a probable square by inspection no matter how Large it is we must see if the last number is one of the foll:
1,4,9,6,5 and it goes on in the reverse sequence as 5,6,9,4,1 and so on and on. Of course this is elementary but few people use this.[/QUOTE]
The number 576 = 24[sup]2[/sup] is a perfect square. According to your last paragraph, it ends in 6, so let's continue. But the second digit from the right is not on your list. So it appears that something is wrong.
[QUOTE=mfgoode]P.S. BTW How far do tables on squares go? a million? or 10 million of them? This will be easy by your programs to tabulate them. I mean somehing like Lehmer's table of primes which can be used as a ready reference for those who need it
As You are onto sequences, perhaps you could add these on as part of your
web site and repertoire. Only a suggestion.[/QUOTE]
Using a computer you can find if a number of 10000 digits is a perfect square in a fraction of second (using clever programming, of course). So it is pointless to store tables of squares, unless they also have other interesting properties.

Squares and Factorisation
 

[QUOTE=alpertron;84972]The number 576 = 24[sup]2[/sup] is a perfect square. According to your last paragraph, it ends in 6, so let's continue. But the second digit from the right is not on your list. So it appears that something is wrong.

Using a computer you can find if a number of 10000 digits is a perfect square in a fraction of second (using clever programming, of course). So it is pointless to store tables of squares, unless they also have other interesting properties.[/QUOTE]
:smile:
Squares and factorisation.
The last digits 0, I , 4 , 5 , 6, 9 are okay for a cursory check, but not enough to be reasonably sure of the number being a perfect square.
With some factorisation methods and problems in number theory, it is of importance to decide quickly whether a number can be a perfect square.
I therefore give the last two digits in a square number.
These digits are limited to the following 22 possibilities.
00 21 41 64 89
01 24 44 69 96
04 25 49 76
09 29 56 81
16 36 61 84

Another useful observation is that when the smallest prime factor p of a number is found to be greater than the cube root [3/rt.] of n the other factor m in n=p*m must be prime.
Thus if m = a*b were composite both a and b would exceed the cube root n and one would obtain the contradiction
n=p*.a*.b > (3/rt).* n *(3/rt). n *(3/rt)*.n = n

Eg: Find the prime factorisation of n = 377,161.
Find sq.rt. n < 614.
The smallest factor is p=137 and n=137*2753.
But (3/rt). n < 73.
Hence 2753 is a prime.
This may be old hat to you but I’m putting it down for the record
Mally :coffee: