AMC 10 · 2015 · #23
Grade 6 arithmeticPick an answer.
AMC 10 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
There is no neat formula spitting out the answer, but each candidate n is easy to test once we know how to count trailing zeros, so tool #6 (Guess and Check) drives the search: try n=5,6,7,… in order and keep the ones that work. Before testing, tool #16 (Change Focus) makes each test cheap — instead of chasing factors of 10, we count only factors of 5, since fives are rarer than twos and so decide the number of zeros. Tool #2 (Make a Systematic List) keeps the trials organized in a small table of n, k, and the zeros of (2n)!, so the four winners stand out without missing any.
Count only the fives
A zero needs a paired 2 and 5, and 5's run out before 2's, so a factorial's zero-count is exactly how many 5's it holds.
Zeros need a 2 and a 5 paired up, and fives run out first, so the fives alone set the count.
4.OA.B.4Count The ComplementA way to count the fives
Count factors of 5 in m! by adding the multiples of 5 up to m, then the multiples of 25, and so on.
Each higher power of 5 contributes one more factor, so you add a fresh layer of count for 25, 125, and so on.
6.NS.B.2Make A Systematic ListWrite the requirement
Write Z(m) for the zero-count of m!; then the test on each n is simply whether Z(2n) equals three times Z(n).
Naming the zero-count Z turns the wordy condition into one tidy equation you can test number by number.
6.EE.B.6Guess And CheckTest n from 5 to 9
n=5,6,7 give only 2 zeros in (2n)! (need 3), but n=8 and n=9 hit Z(16)=Z(18)=3, so 8 and 9 are the first two winners.
Doubling n across a multiple of 5 (here 16 and 18 clear 15) suddenly adds a third five, which is exactly what tripling the count needs.
6.NS.B.2Guess And CheckTest n from 10 to 14
n=10,11,12 reach only 4 zeros (need 6), but at 2n=26,28 the number 25 adds a bonus 5, so Z(26)=Z(28)=6 and 13,14 are the next winners.
Once 2n reaches 26, the number 25 donates a bonus five, lifting the zero-count from 4 up to the needed 6.
6.NS.B.2Guess And CheckAdd and take the digit sum
The four winners 8,9,13,14 add to s=44, and its digits give 4+4=8.
Adding the four winners and splitting the total into its digits is the last small step once the search is done.
4.NBT.B.4Guess And CheckTrailing zeros come only from factors of 5, so count the fives, then just try n in order and keep the ones where (2n)! has three times as many.
- Count only the fives
- A way to count the fives
- Write the requirement
- Test n from 5 to 9
- Test n from 10 to 14
- Add and take the digit sum