1992 AIME Problems/Problem 6
Problem
For how many pairs of consecutive integers in is no carrying required when the two integers are added?
Solution
Solution 1
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 .
Solution 2
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.
Solution 3
There are 3 forms possible. : , : , : , : --- Thus, since there should be no carrying, in only integers to is possible Therefore, the answer is
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