When you multiply 102564 by 4, its rightmost digit moves to the left and the other digits move one space to the right (4 × 102564 = 410256). Find the smallest number that has this same property when multiplied by N (where N is each integer from 2 to 9.) Some answers are surprisingly long.

Source: Based on Macalaster College Problem of the Week #1013.

Solutions were received from Philippe Fondanaiche, Kirk Bresniker, Denis Borris, Joseph DeVincentis, and Dan Chirica. Philippe pointed out that I had a previous puzzle similar to this on February 3, 1999, and its solution lists all the solutions to this one. Sorry 'bout that. For this week, though, I think Joseph's solution is a good summary:

So we need to solve: n(10a+b) = (10^k)b + a where b is an integer 1 to 9 and a is an integer in [10^(k-1),10^k) and k is the smallest integer where it has a solution, for each n from 2 to 9. Solve for a: a = b(10^k - n)/(10n-1) So we need to find the smallest k such that 10^k - n is divisible by 10n-1, or at least by a large factor of 10n-1 in those cases (39, 49, 69) where it is not prime. This will generally be an order of magnitude less than 10^k, so then we need to find the smallest b so make a be at least 10^(k-1). For n=2, we have to go up to k=17 to find a/b = 99999999999999998/19 = 5263157894736842. Then b=2, a=10526315789473684, and 2 * 105263157894736842 = 210526315789473684. For n=3, we have to go up to k=27 to find a/b = 999999999999999999999999997/29 = 34482758620689655172413793. Then b=3, a = 103448275862068965517241379, and 3 * 1034482758620689655172413793 = 3103448275862068965517241379 For n=4, the denominator 39 = 3 * 13 so we may be able to get away with 10^k - 4 only being a multiple of 13 if b is a multiple of 3. But since 10^k - 4 = 999...9996, it is a multiple of 3 anyway. We find a/b = 99996/39 = 2564, b = 4, a = 102564, and 4 * 102564 = 410256 as in the example. For n=5, the denominator 49 = 7*7 so we may be able to get away with 10^k - 5 is only divisible once by 7. Indeed, this turns up the short solution a/b = 99995/49 = 14285/7, b = 7, and 5 * 142857 = 714285. For n=6, we have to go up to k=57 to find a/b = 999999999999999999999999999999999999999999999999999999994/59 = 16949152542372881355932203389830508474576271186440677966. Then b=6, and 6 * 1016949152542372881355932203389830508474576271186440677966 = 6101694915254237288135593220338983050847457627118644067796. For n=7, the denominator 69 = 3 * 23, but as with the n=4 case, 10^k - 7 is divisible by 3 so this doesn't reduce the size of the solution. We need k=21, a/b = 999999999999999999993/69 = 14492753623188405797L. Then b = 7, and 7 * 1014492753623188405797 = 7101449275362318840579. For n=8, k=12, a/b = 999999999992/79 = 12658227848. So b=8, and 8 * 1012658227848 = 8101265822784. For n=9, k=43, a/b = 9999999999999999999999999999999999999999991/89 = 112359550561797752808988764044943820224719. b = 9, and 9 * 10112359550561797752808988764044943820224719 = 91011235955056179775280898876404494382022471.

Mail to Ken