Problem: (1978 IMO – #1) We consider all the pairs  of integer numbers, so that  and so that their last three digits from the decimal writing of  are the same with those three last digits from the decimal writing of  . Find all the pairs with these properties so that the sum  is minimum.
Solution: Basically it’s asking us to find the smallest  and  such that  or  . By the looks of this problem, Euler’s Totient Theorem might come in handy again. But we have a small problem –  .
So we try dividing out all the factors of  in  .
By the Totient Theorem, we find  and that  is also the smallest power for which this is true. This means  for some positive integer  .
Hence and differ by at least . Suppose  .
Then  . Since the second part is odd, all the powers of two, there must be at least three of them, come from  . Hence  . But since we know  , we have  divisible by both  and  , which means it’s divisible by , so  is the smallest possible value that works, giving  .
Therefore, our smallest  . QED.
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Comment: Kind of a strange problem, with the minimum value, reminiscent of many AIME problems. Ultimately, not too difficult, though.
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Practice Problem: Find the value of  .
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