# Thread: the largest integer that divides all the number of the form ABCABC

1. ## the largest integer that divides all the number of the form ABCABC

Let ABC be a 3-digit number such that its digits A, B, and C form
an arithmetic sequence. The largest integer that divides all numbers of
the form ABCABC is ______ ?

thanks

2. Originally Posted by rcs
Let ABC be a 3-digit number such that its digits A, B, and C form
an arithmetic sequence. The largest integer that divides all numbers of
the form ABCABC is ______ ?

thanks
So what ideas have you had for tackling this?

Is there some related material you have covered in class?

(As a start you will observe that ABCABC is always divisible by 1001)

CB

3. some of the problems i posted are of course related in the class... i have some problems that i couldn't solve that is why i need help from here... some problems i can solve is of course not posted here because i know how. Only those i can hardly solve... i thought the more problems posted the better. im sorry sir if i have misconstrued something in here.

thanks a lot. God Bless

4. Originally Posted by rcs
some of the problems i posted are of course related in the class... i have some problems that i couldn't solve that is why i need help from here... some problems i can solve is of course not posted here because i know how. Only those i can hardly solve... i thought the more problems posted the better. im sorry sir if i have misconstrued something in here.

thanks a lot. God Bless
You are expected to show what you have tried or explain what exactly is the problem you are having with a particular question. Then we can help you to a solution that you will have contributed to yourself.

We are not here to do your homework, assignments, take-out exams or provide solutions for a solutions manual for you despite what some posters here may seem to think.

CB

5. yeah sir i know, and i understand that from the very first time i joined here. i cant seem think on what to do, that is why i tried to post it to have a little idea from yours or anybody from the helpers here.

Good Day!

6. Originally Posted by rcs
yeah sir i know, and i understand that from the very first time i joined here. i cant seem think on what to do, that is why i tried to post it to have a little idea from yours or anybody from the helpers here.

Good Day!
You have been told that every number of the form can be written $1001 \times ABC$ where the digits $A, B, C$ are is arithmetic progression. Now try a few examples, what is the greatest common factor that you find?

CB

7. 1001 x 234 = 234234 , A = 2, B=3, C = 4
1001 x ABC = ABCABC

thanks

8. Originally Posted by rcs
1001 x 234 = 234234 , A = 2, B=3, C = 4
1001 x ABC = ABCABC

thanks
So you have now narrowed the answer down the largest possible factor of such a number to one of $1001\times a_i$ where $a_i$ is a factor of $234$, but $234$ is not particularly convenient for this work as it has rather more factors than is convenient.

If instead we work with $A=1, B=2, C=3$ , $123=41\times 3$.

So the greatest common factor of numbers of the type being considered is one of $1001, 3003, 4141, 123123$.

So Now try $A=3, B=5, C=5$, $345=3\times 5 \times 23$, combining this with the previous result tells us the greatest common factor is one of $1001$ and $3003$.

To complete the solution it is sufficient to decide if a number of the form $100\times A+10\times B+C$ with $A, B, C$ being in arithmetic progressions (and single digit numbers) is always divisible by $3$. So what rules do you know for testing divisibility by $3$?

CB