1992 AIME Problems/Problem 6
For how many pairs of consecutive integers in is no carrying required when the two integers are added?
For one such pair of consecutive integers, let the smaller integer be where and are digits from through
We wish to count the ordered triples By casework, we consider all possible forms of the larger integer, as shown below. Together, the answer is
Consider what carrying means: If carrying is needed to add two numbers with digits and , then or or . 6. Consider . has no carry if . This gives possible solutions.
With , there obviously must be a carry. Consider . have no carry. This gives possible solutions. Considering , have no carry. Thus, the solution is .
Consider the ordered pair where and are digits. We are trying to find all ordered pairs where does not require carrying. For the addition to require no carrying, , so unless ends in , which we will address later. Clearly, if , then adding will require no carrying. We have possibilities for the value of , for , and for , giving a total of , but we are not done yet.
We now have to consider the cases where , specifically when . We can see that , and all work, giving a grand total of ordered pairs.
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