## CSMC 2021 Part B - Question 2, CEMC UWaterloo

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#### Difficulty: 6

#### This problem is tagged with csmc, csmc21, highschool.

**(Canadian Senior Mathematics Contest 2021, Part B, Question 2, CEMC - UWaterloo)**

**Adapted from original statement.**

If $n$ is a positive integer, a *Leistra sequence* is a sequence $a_1,a_2,a_3, \dots, a_{n-1},a_n$ with $n$ terms with the following properties:

(P1) Each term $a_1,a_2,a_3, \dots, a_{n-1},a_n$ is an even positive integer.

(P2) Each term $a_1,a_2,a_3, \dots, a_{n-1},a_n$ is obtained by dividing the previous term in the sequence by an integer between $10$ and $50$, inclusive. (For a specific sequence, the divisors used do not all have to be the same.)

(P3) There is no integer $m$ between $10$ and $50$, inclusive, for which $\frac{a_n}{m}$ is an even integer.

The image below shows some examples:

a) Determine all Leistra sequences with $a_1 = 216$.

b) How many Leistra sequences have $a_1 = 2 \times 3^{50}$?

c) How many Leistra sequences have $a_1 = 2^2 \times 3^{50}$?

d) Determine the number of Leistra sequences with $a_1 = 2^3 \times 3^{50}$.

*Answer Submission Note(s)*

For the first part, your answer must have all possible Leistra sequences (a sequence with only two integers $x$ and $y$ should be represented as "x,y" - comma-separated, no spaces). Your Leistra sequences must be ordered in lexicographic order. Separate individual sequences with a single space. For the other three parts, your answer must have a single integer each. Separate answers for all parts with a single space when submitting.

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