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jmc

number theory intermediate

Problem

When the single digit is written at the end of the positive two-digit integer , with tens digit and ones digit , the resulting positive three-digit integer is 619 more than the original integer . What is the value of the three-digit integer ?
Solution
We are given . Examining hundreds digits, we know that is or . Examining tens digits, on the right side we cannot have carrying into the hundreds digit, so , and thus is or . However, we see that the sum on the right side must have carrying into the tens digit, so . Finally, is trivial. Thus .
Final answer
687