2003 AMC 10B Problems/Problem 25
- 1 Problem
- 2 Solution
- 3 See also
How many distinct four-digit numbers are divisible by and have as their last two digits?
Solution 1 (Slow)
To test if a number is divisible by three, you add up the digits of the number. If the sum is divisible by three, then the original number is a multiple of three. If the sum is too large, you can repeat the process until you can tell whether it is a multiple of three. This is the basis for the solution. Since the last two digits are , the sum of the digits is (therefore it is not divisible by three). However certain numbers can be added to make the sum of the digits a multiple of three.
, and so on.
However since the largest four-digit number ending with is , the maximum sum is
Using that process we can fairly quickly compile a list of the sum of the first two digits of the number.
Now we find all the two-digit numbers that have any of the sums shown above. We can do this by listing all the two digit numbers in separate cases.
And finally, we add the number of elements in each set.
Solution 2 (Medium)
A number divisible by has all its digits add to a multiple of The last two digits are and and add up to Therefore the first two digits must add up to digits (including ) are are and are The following combinations are equivalent to :
Let the first term in each combination represent the thousands digit and the second term represent the hundreds digit. We can use this to find the total amount of four-digit numbers.
Solution 3 (Fast)
We have the following: and . Then for some integers and . Taking mod 3 gives: so for some integer . But so . Bounding this gives us: so . Dividing by gives so . This gives
Solution 4 (Extremely Fast)
The number is in the form
Notice that the number is divisible by if the sum of the digits of is divisible by
Our first number that is divisible by is , next is .. Notice goes from
Hence, there are distinct four digit numbers.
Solution 5 (Even Faster than Extremely Fast)
Following the form in Solution 4, notice that to satisfy our condition. Choose a value of from 1 to 9. For , there are exactly 4 values of . For the remaining 6 digits, there are 3 choices for y. So our answer is .
Solution 6 (Even Faster than Even Faster than Extremely Fast)
There are possible values for the first two digits. of them yield a multiple of , so the answer is
Solution 7 (Easy to understand)
Denote the four digit number by . Then it is well known the sum of the digits of an integer determine whether or not they are divisible by . Thus we want .
Now we have that when this is the same case as as . Hence if we count the above number of solutions and multiply by (as ), we are home free. There are solutions. Multiplying this by yields .
Solution 8 (Very very inefficient and slow)
Don't do this on a test unless you have lots of time left.
Pounding out all of the possible numbers and counting them, we get
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