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

please give me the complete solution of this problem.....

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- Nov 29th 2010, 07:08 PMjpmath2010ABC is a 3-digit number such that its digits A, B and C form an arithmetic sequence..
Let ABC be a 3-digit number such that its digits A, B, and C form and arithmetic sequence. The largest integer that divides all numbers of form ABCABC is.....?

please give me the complete solution of this problem..... - Nov 29th 2010, 08:50 PMSoroban
Hello, jpmath2010!

Is that the exact wording of the problem?

As stated. it's a silly problem . . .

Quote:

. . .

The largest integer that divides is . . .

- Nov 29th 2010, 10:26 PMelemental
.

Soroban, he is asking what number divides ALL numbers of this form. For example,

the GCD of 123123 and 135135 is 3003. The GCD of 147147 and 963963 is also 3003. - Nov 30th 2010, 04:35 AMjpmath2010
thank you soroban for your effort....Any way the problem i have posted is come the Mathematics Olympiad.That is the exact problem. would you please give me the exact solution of that problem.

- Nov 30th 2010, 09:52 AMSoroban
Hello, jpmath2010!

Now I get it . . .

Quote:

Let be a 3-digit number such that its digits form and arithmetic sequence.

Find the largest integer that divides all numbers of form

Let = hundreds digit of the three-digit number,

. . and be the common difference of the arithmetic sequence.

Then: .

Hence: .

Therefore, the largest divisor is

- Nov 30th 2010, 05:08 PMjpmath2010
wow thats great....i really thank you so much....have a nice day and God bless...