# Special Factors of 3-Digit Numbers

phgao
Posers and Puzzles 22 Feb '06 13:13
1. 22 Feb '06 13:131 edit
Find all positive integers n (n will be called special)
such that if n divides a three digit number say ABC where A = hundreds, B = tens, C = ones. Then n also divides BCA and CAB.

It's trickier than you think ...
2. 22 Feb '06 13:16
Originally posted by phgao
Find all positive integers n (n will be called special)
such that if n divides a three digit number say ABC where A = hundreds, B = tens, C = ones. Then n also divides BCA and CAB.

It's trickier than you think ...
I'd say 3 is one of them.
3. 22 Feb '06 13:21
Originally posted by fetofs
I'd say 3 is one of them.
Correct as if the ABC divides 3, then moving around the digits the sum of ABC is still the same.

There are more...
4. 22 Feb '06 13:211 edit
Originally posted by fetofs
I'd say 3 is one of them.
5. 22 Feb '06 13:38
Originally posted by phgao
Correct as if the ABC divides 3, then moving around the digits the sum of ABC is still the same.

There are more...
9 as well.
6. 22 Feb '06 13:45
Originally posted by fetofs
9 as well.
yea keep going...
7. 22 Feb '06 14:07
isnt 1 also a solution? that would make it 1,3,9 so far which is 3^0, 3^1 and 3^2.....maybe 3^3 (=27) also works then?

432:27 = 16
324:27 = 12
243:27 = 9

621:27= 23
216:27= 8
162:27= 6

you meant it like that right?

for 3^4=81 i can find only pairs of two 3-digit-numbers though. i guess thats because 81 is a too large number compared to 3-digit numbers.
8. BigDoggProblem
22 Feb '06 23:552 edits
1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 15, 18, 21, 24, 27, 37, 54, 74, 111, 148, 185, 222, 259, 296, 333, 444, 555, 666, 777, 888, 999

Next Problem!

Edit: The stipulation doesn't rule out the possibility that A=B=C and thus n=2 is a solution because it goes into 222 evenly, and switching digits has no effect.

Edit2: Even disallowing A=B=C still leaves n=2 as a solution, because it goes into 246, 462, and 624 evenly.
9. XanthosNZ
Cancerous Bus Crash
23 Feb '06 00:591 edit
Originally posted by BigDoggProblem
1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 15, 18, 21, 24, 27, 37, 54, 74, 111, 148, 185, 222, 259, 296, 333, 444, 555, 666, 777, 888, 999

Next Problem!

Edit: The stipulation doesn't rule out the possibility that A=B=C and thus n=2 is a solution because it goes into 222 evenly, and switching digits has no effect.

Edit2: Even disallowing A=B=C still leaves n=2 as a solution, because it goes into 246, 462, and 624 evenly.
I believe the problem asks for numbers n such that if n is a factor of ABC then it will also be a factor of BCA and CAB.

The reason we can show that 3 satisfies this is because of the division check where if A+B+C = 3 or a number divisible by 3 then ABC is divisible by 3. The rearrangements have the same sum so this works. 9 works by the same principle. 1 works because it divides all numbers.

2 doesn't work. A counterexample is 136. 2 is a factor of this number but not 361 or 613.

EDIT: I'm writing some code that will find all such numbers however I need a clarification, what happens with numbers with a zero in them? In one rearrangement they will have a leading zero. Is this a problem?
10. BigDoggProblem
23 Feb '06 01:401 edit
Originally posted by XanthosNZ
I believe the problem asks for numbers n such that if n is a factor of ABC then it will also be a factor of BCA and CAB.

The reason we can show that 3 satisfies this is because of the division check where if A+B+C = 3 or a number divisible by 3 then ABC is divisible by 3. The rearrangements have the same sum so this works. 9 works by the same principle. rs with a zero in them? In one rearrangement they will have a leading zero. Is this a problem?
I believe your interpretation of the stipulation is indeed what the original poster intended. Thanks for clearing that up.

My new adjusted list is:

1, 3, 9, 27, 37, 111, 222, 333, 444, 555, 666, 777, 888, 999

Edit: In my code, I ruled out any three digit numbers that included leading zeros.
11. TheMaster37
Kupikupopo!
23 Feb '06 07:26
Originally posted by BigDoggProblem
I believe your interpretation of the stipulation is indeed what the original poster intended. Thanks for clearing that up.

My new adjusted list is:

1, 3, 9, 27, 37, 111, 222, 333, 444, 555, 666, 777, 888, 999

Edit: In my code, I ruled out any three digit numbers that included leading zeros.
148 : 37 = 38

Though 184 isn't a multiple of 37 (185 is).

Try harder Doggie ðŸ™‚
12. XanthosNZ
Cancerous Bus Crash
23 Feb '06 07:40
Originally posted by TheMaster37
148 : 37 = 38

Though 184 isn't a multiple of 37 (185 is).

Try harder Doggie ðŸ™‚
But 184 isn't one of the arrangements that 37 must divide (those are 481 and 814 both of which are multiples of 37).
13. BigDoggProblem
23 Feb '06 07:46
Originally posted by TheMaster37
148 : 37 = 38

Though 184 isn't a multiple of 37 (185 is).

Try harder Doggie ðŸ™‚
You have a minor problem. You did not change the position of the "1".

Please remember, when dealing with me, you are no longer TheMaster, but the student. Call me 'doggie' one more time and I will crudely disembowel you before all the forum audience.
14. TheMaster37
Kupikupopo!
23 Feb '06 11:451 edit
Originally posted by BigDoggProblem
You have a minor problem. You did not change the position of the "1".

Please remember, when dealing with me, you are no longer TheMaster, but the student. Call me 'doggie' one more time and I will crudely disembowel you before all the forum audience.
My bad BigProblem, I do apologise :p

As long as my nickname is TheMaster, I will remain TheMaster. That nickname, however, in no way represents my level of expertise; I am entitled to my share of flaws and mistakes.

I do find it hilarious that you of all people would name anyone a student.

EDIT; Might be an idea to prove that your claim is correct. Are those the only numbers?
15. 23 Feb '06 15:24
Originally posted by BigDoggProblem
Please remember, when dealing with me, you are no longer TheMaster, but the student. Call me 'doggie' one more time and I will crudely disembowel you before all the forum audience.
cant wait to see that ðŸ˜€

So 1,3,9,27 fit into a system (3^x)...so do 111, 222 etc.....but what about 37? thats a funny number there, but it surely works also.