1992 AIME Problems/Problem 6
Contents
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.
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