2006 AIME II Problems/Problem 7
Problem
Find the number of ordered pairs of positive integers such that and neither nor has a zero digit.
Solution
Solution 1
There are numbers up to 1000 that have 0 as their units digit. All of the other excluded possibilities are when or have a 0 in the tens digit, and since the equation is symmetric, we will just count when has a 0 in the tens digit and multiply by 2 (notice that the only time both and can have a 0 in the tens digit is when they are divisible by 100, which falls into the above category, so we do not have to worry about overcounting).
Excluding the numbers divisible by 100, which were counted already, there are numbers in every hundred numbers that have a tens digit of 0 (this is true from 100 to 900), totaling such numbers; considering also and we have . Therefore, there are such ordered pairs.
Solution 2
Let and be 3 digit numbers:
cde +fgh ---- 1000
and must add up to , and must add up to , and and must add up to . Since none of the digits can be 0, there are possibilites if both numbers are three digits.
There are two other scenarios. and can be a three digit number and a two digit number, or a three digit number and a one digit number. For the first scenario, there are possibilities (the two accounting for whether or has three digits) and for the second case there are possibilities. Thus, thus total possibilities for is .
Solution 3
We first must notice that we can find all the possible values of between and and then double that result.
When there are possible solution for so that neither nor has a zero in it, counting through , through , ..., through . When there are possible solution for a so that neither a nor b has a zero in it, counting through , through , ..., through . This can clearly be extended to where is an integer and . Thus for there are = possible values of .
Thus when there are possible values of and .
Doubling this yields .
See also
2006 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
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