HELP NEEDED PLEASE

[scottishstudent]

How many positive integers less than or equal to 123456 have a multiple that consists of all the same digits? eg 101 works since 101 * 11 = 1111 which consists of only 1s)

 

[Steven G]

Nice problem. Where are you stuck? It is hard to help you if you do not show us any work.

 

[Steven G]

Have you considered the numbers 1, 11, 111, ..., 111111, 2, 22, ... ,22222,..., 99999?

 

[scottishstudent]

Have you considered the numbers 1, 11, 111, ..., 111111, 2, 22, ... ,22222,..., 99999?

Thank you for the quick response? we have just started this topic so I’m having difficulty working it out

 

[Steven G]

Isn't it clear that the numbers I listed must be the products?

 

[scottishstudent]

thank you! yeah I think I understand that part, there are several more like these and if I could just understand how to work out the answer to this one I can do the others.

 

[pka]

How many positive integers less than or equal to 123456 have a multiple that consists of all the same digits? eg 101 works since 101 * 11 = 1111 which consists of only 1s)

If we consider the factorization of \(444444=2^2\cdot3\cdot7\cdot11\cdot13\cdot37\) SEE HERE
That tells us that there are \(96\) divisors of \(444,444\); each divisor gives a pair like \(\{308,1443\}\)
So there are 48 such pairs.
We almost need a programmer to test all.

 

[Steven G]

thank you! yeah I think I understand that part, there are several more like these and if I could just understand how to work out the answer to this one I can do the others.

You say that there are several more like these. Can you please list some?

 

[pka]

thank you! yeah I think I understand that part, there are several more like these and if I could just understand how to work out the answer to this one I can do the others.

thank you! yeah I think I understand that part, there are several more like these and if I could just understand how to work out the answer to this one I can do the others.

You say that there are several more like these. Can you please list some?

I am of the opinion that whoever wrote to original post had no idea as to how complicated s/he was making the problem.
Had the count been limited to six-digit numbers no larger than \(666666\) then that would have limited it to a doable by hand problem.
But as written there is such a huge collection of possibles. Thus I think it was written by an amateur. If I am wrong about that I apologize. In that case it should have been made abundantly that this was a programming question. In either case, it is a faulty question.