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
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
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