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This problem trains for: AMC-8, AMC-10, AMC-12, AIME, Math Kangaroo 7-8, Math Kangaroo 9-10.

If a and m are two different unknown digits of a 5-digit positive integer that is divisible by 17 and is of the form amama, how many different values of the sum a+m are there?

Write the number in a partially expanded form:

equation

and factor the numbers into primes:

equation

Now cast out the seventeens:

equation

or, equivalently, use congruences:

equation

to show that, for the number of the form amama to be a multiple of 17 we have to have:

equation

This is a linear Diophantine equation. It normally has an infinity of solutions over Z2, but we will use additional information to limit the number of solutions as follows:

Since the maximum value of any digit in the decimal system is 9, we have to look only for multiples of 17 that are larger than 3 and smaller than 90.

equation

For k=1, we have:

equation

equation

For k=2, we have:

equation

equation

as well as:

equation

with the solution:

equation

For k=3, we have:

equation

with the solution:

equation

equation

For k=4, we have:

equation

equation

There are five distinct solutions: 12121, 24242, 36363, 48484, 91919 and there are five distinct sums a+m: 3, 6, 9, 10, 12.