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Difference between revisions of "2006 AIME II Problems/Problem 7"

Problem

Find the number of ordered pairs of positive integers $(a,b)$ such that $a+b=1000$ and neither $a$ nor $b$ has a zero digit.

Solution

Solution 1

There are $\left\lfloor\frac{999}{10}\right\rfloor = 99$ numbers up to 1000 that have 0 as their units digit. All of the other excluded possibilities are when $a$ or $b$ have a 0 in the tens digit, and since the equation is symmetric, we will just count when $a$ has a 0 in the tens digit and multiply by 2 (notice that the only time both $a$ and $b$ 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 $9$ numbers in every hundred numbers that have a tens digit of 0 (this is true from 100 to 900), totaling $9 \cdot 9 = 81$ such numbers; considering $b$ also and we have $81 \cdot 2 = 162$. Therefore, there are $999 - (99 + 162) = \boxed{738}$ such ordered pairs.

Solution 2

Let $a = \overline{cde}$ and $b = \overline{fgh}$ be 3 digit numbers:

cde
+fgh
----
1000 $e$ and $h$ must add up to $10$, $d$ and $g$ must add up to $9$, and $c$ and $f$ must add up to $9$. Since none of the digits can be 0, there are $9 \times 8 \times 8=576$ possibilites if both numbers are three digits.

There are two other scenarios. $a$ and $b$ 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 $9 \times 8 \times 2=144$ possibilities (the two accounting for whether $a$ or $b$ has three digits) and for the second case there are $9 \times 2=18$ possibilities. Thus, thus total possibilities for $(a,b)$ is $576+144+18=738$.

Solution 3

We first must notice that we can find all the possible values of a between 1 and 500 and then double that result.

When 1 < a < 100 there are 9 X 9 = 81 possible solution for a so that neither a nor b has a zero in it, counting 1 through 9, 11 through 19 ... 81 through 89. When 100 < a < 200 there are 9 X 8 =72 possible solution for a so that neither a nor b has a zero in it, counting 111 through 119, 121 through 129 ... 181 through 189. This can clearly be extended to 100k < a < 100(k+1) where k is an integer and 0 < k < 9. Thus for 100 < a < 500 there are 72 X 4 = 288 possible values of a.

Thus when 1 < a < 500 there are 288 + 81 = 369 possible values of a and b.

Doubling this yields the 738.