11 replies to this topic

[bushindo]

This is an extension of Prime's excellent puzzle One's Divisibility

 

What is the smallest 101-plus-digit number consisting of only 4's, 5's, and 7's that is divisible by '7777.....777' (one hundred 7's)?

 

The solution *must contain* at least one 4, one 5, and one 7.

 

EDIT: clarified the properties of the required solution

[Rob_Gandy]

I'm assuming it has to have atleast one 4, 5 and 7?

[bushindo]

I'm assuming it has to have atleast one 4, 5 and 7?

 

That is correct. I'll edit the OP to clarify that point. Thanks.

[Prime]

This looks harder than the puzzle I posted.

Spoiler for Can't find it

 divisibility.pdf   26.54K   30 downloads

[Prime]

Found a roundabout way to display formatted text inside a BD post. Same thing as above, but visible.

Spoiler for can't find it, but here is a way

[bushindo]

Found a roundabout way to display formatted text inside a BD post. Same thing as above, but visible.

Spoiler for can't find it, but here is a way

divisibility.gif

You are approaching this from the correct angle

Spoiler for

Breaking the 100 7s into 7 * 100 1s is the right first step. As to the analysis above, I recommend trying some small case like 11 or 111. It shouldn't be hard to find the solution in those cases.

[Prime]

Spoiler for here's one big number

I found this interesting 102-digit number: 490000...000077 (“49”, followed by 98 zeros, followed by “77”.) If you multiply that number by 111...111 (one hundred 1-s), the resulting 201-digit number: 54444...4447555...55547 (“5”, followed by 99 “4-s”, followed by “7”, followed by 98 “5-s”, followed by “47”) meets the conditions set forth in the OP, except I cannot guaranty it is the smallest number that does. (I did not find any simple combination of "1-s" and "0-s", which multiplied by all "7-s" would yeild "4-s", "5-s", and "7-s", while avoiding "8-s".)

On a side note. While playing with these numbers I found a simple yet curious divisibility rule, of which I was not aware before. I should try and construct a problem based on it.

[bushindo]

Spoiler for here s one big number

I found this interesting 102-digit number: 490000...000077 (“49”, followed by 98 zeros, followed by “77”.) If you multiply that number by 111...111 (one hundred 1-s), the resulting 201-digit number: 54444...4447555...55547 (“5”, followed by 99 “4-s”, followed by “7”, followed by 98 “5-s”, followed by “47”) meets the conditions set forth in the OP, except I cannot guaranty it is the smallest number that does. (I did not find any simple combination of "1-s" and "0-s", which multiplied by all "7-s" would yeild "4-s", "5-s", and "7-s", while avoiding "8-s".)

On a side note. While playing with these numbers I found a simple yet curious divisibility rule, of which I was not aware before. I should try and construct a problem based on it.

Excellent work so far. Sorry for not responding earlier. I was on a vacation and didn't have access to a computer

Spoiler for

The number above indeed satisfies the conditions giving in the OP, but there is a smaller number. The number I have in mind is a 201-digit number, but the 201-th (leftmost) digit is a 4.

I'm interested in seeing a puzzle based on the divisibility rule you found. For what it is worth, I also based this puzzle on a divisibility rule as well.

[bushindo]

Spoiler for here s one big number

I found this interesting 102-digit number: 490000...000077 (“49”, followed by 98 zeros, followed by “77”.) If you multiply that number by 111...111 (one hundred 1-s), the resulting 201-digit number: 54444...4447555...55547 (“5”, followed by 99 “4-s”, followed by “7”, followed by 98 “5-s”, followed by “47”) meets the conditions set forth in the OP, except I cannot guaranty it is the smallest number that does. (I did not find any simple combination of "1-s" and "0-s", which multiplied by all "7-s" would yeild "4-s", "5-s", and "7-s", while avoiding "8-s".)

On a side note. While playing with these numbers I found a simple yet curious divisibility rule, of which I was not aware before. I should try and construct a problem based on it.

Excellent work so far. Sorry for not responding earlier. I was on a vacation and didn't have access to a computer

Spoiler for

The number above indeed satisfies the conditions giving in the OP, but there is a smaller number. The number I have in mind is a 201-digit number, but the 201-th (leftmost) digit is a 4.

I'm interested in seeing a puzzle based on the divisibility rule you found. For what it is worth, I also based this puzzle on a divisibility rule as well.

 

Solution to the puzzle

Spoiler for

 

If we multiply 100 7's with the following number

 

 58 571 428 571 428 571 428 571 428 571 428 571 428 571 428 
571 428 571 428 571 428 571 428 571 428 571 428 571 428 571 
428 571 428 602

 

We would get the following 201-digit number

 

455 555 555 555 555 555 555 555 555 555 555 555 555 555 555 
555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 
555 555 555 747 777 777 777 777 777 777 777 777 777 777 777 
777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 
777 777 777 777 777 777 754

 

If you look closely at the number above, you might be able to see the divisibility rule used in the construction of this puzzle.

[Prime]

Indeed, this number is a bit smaller than the one I found.

Spoiler for another perspective

Perhaps, it is easier to evaluate as 410000...000214: "41", 97 zeros, and "214" (which is, clearly, divisible by 7) multiplied by 100 "1"s.

But where is the proof that it is the smallest number that meets  the conditions?

Edited by Prime, 09 March 2013 - 08:17 AM.