# 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 $\displaystyle 1001 \times ABC$ where the digits $\displaystyle A, B, C$ 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 $\displaystyle 1001\times a_i$ where $\displaystyle $$a_i is a factor of \displaystyle 234, but \displaystyle 234 is not particularly convenient for this work as it has rather more factors than is convenient. If instead we work with \displaystyle A=1, B=2, C=3 , \displaystyle 123=41\times 3. So the greatest common factor of numbers of the type being considered is one of \displaystyle 1001, 3003, 4141, 123123. So Now try \displaystyle A=3, B=5, C=5, \displaystyle 345=3\times 5 \times 23, combining this with the previous result tells us the greatest common factor is one of \displaystyle 1001 and \displaystyle 3003. To complete the solution it is sufficient to decide if a number of the form \displaystyle 100\times A+10\times B+C with \displaystyle A, B, C being in arithmetic progressions (and single digit numbers) is always divisible by \displaystyle$$3$. So what rules do you know for testing divisibility by $\displaystyle$$3$?

CB