AMC 10 · 2017 · #16
Grade 7 probabilityPick an answer.
Tool #16 (Change Focus): a divisor's oddness depends only on how many 2's it contains, so I ignore every other prime and watch only the power of 2. Tool #7 (Identify Subproblems): the one number I really need is how many 2's hide inside 21!, a self-contained counting job. Tool #4 (Introduce a Variable): I let the exponent of 2 in a divisor be a variable that ranges over a clear list of values, turning 'odd or not' into a clean count.
Oddness depends only on 2's
Odd or even depends only on the prime two.
A number is odd only when it has zero factors of 2, so just track the 2's.
4.OA.B.4Change Focus Count The ComplementCount the 2's in 21!
Counting the twos inside gives 18.
Each multiple of 2 donates a 2, each multiple of 4 donates an extra one, and so on.
Counting the twos means counting multiples of two, then of four, then of eight, and so on.
▸ Why?
Every number has one prime recipe, so its contribution of twos is fixed before anything is multiplied.
▸ Why?
A number holding several twos is a multiple of each power of two, so it is counted once per power.
List the choices for a
The exponent of two has 19 choices, zero included.
Counting 0 through 18 inclusive gives 18+1=19 possible powers of 2.
4.OA.A.3Introduce A VariableTurn the count into a probability
Only one of them is odd, so one nineteenth.
Inside every family that shares an odd part, only the a=0 member is odd, so the odds are 1 in 19.
7.SP.C.7Change Focus Count The ComplementA divisor is odd only when it grabs none of the 2's; 21! has 18 twos, so the power of 2 has 19 choices (0 through 18) and just one keeps it odd — probability 1/19, choice (B).
- Oddness depends only on 2's
- Count the 2's in 21!
- List the choices for a
- Turn the count into a probability