Science Quiz / Swiss Bank 3 Maths Logic Puzzle

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My Long Bank Account Number

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How annoying - my Swiss Bank has decided to insist that all account numbers be between 30 and 90 digits and may not start with a zero. I've no idea why they think this will make things more secure - everyone will have to write their numbers down on scraps of paper which may get lost or stolen. But maybe I can work mine out each time I need it. I'll try to find a number such that If I multiply it by 9 I get the same number with the last digit shifted to the front (so that for example 134689 would become 913468). Can you help me work out what my number has to be and how many digits it may have? There's a solution in "Source".
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