Alice, Beth, and Charlene start climbing a staircase, each with her left foot on step 1 and then her right foot on the next consecutive step. Alice climbs one step at a time, Beth two steps at a time, and Charlene three steps at a time. Each lands on the top step with her left foot. Compute the minimum number of stairs in the staircase.

Question

Alice, Beth, and Charlene start climbing a staircase, each with her left foot on step 1 and then her right foot on the next consecutive step. Alice climbs one step at a time, Beth two steps at a time, and Charlene three steps at a time. Each lands on the top step with her left foot. Compute the minimum number of stairs in the staircase.

I'm not sure how to do this. Should i set up equations?

Answer 1

How about making a chart, with 4 columns

# A B C
1 L L L
2 R - -
3 L R -
4 R - R
5 L L -
6 R - -
7 L R L
8 R - -
9 L L -
10R - R
11L R -
12 R - -
13 L L L , AHHH, all left feet on 13th step

or
step # when Alice will be left footed
1 3 5 7 9 11 13 15 ... (every 2nd step)
step # when Beth will be left-footed
1 5 9 13 17 ... ( every 4th step)
step # when Charlene will be left-footed
1 7 13 19 ... (every 6 steps)

first common number is 13

Answer 2

To solve this problem, you can set up an equation based on the given information and then find the least common multiple (LCM) of the step numbers for Alice, Beth, and Charlene. Let's break it down step by step:

1. Let's denote the number of stairs in the staircase as "n".
2. Alice climbs one step at a time, so she will need to take "n" steps to reach the top.
3. Beth climbs two steps at a time, so she will need to take "n/2" steps to reach the top.
4. Charlene climbs three steps at a time, so she will need to take "n/3" steps to reach the top.

Since all three climbers end on the same step, we can set up the following equation:

n = LCM(n, n/2, n/3)

Now, let's solve this equation to find the minimum number of stairs in the staircase:

1. Find the LCM of n, n/2, and n/3. LCM(n, n/2, n/3) = LCM(n, n/2) = LCM(n, n/3).

2. To find the LCM of two numbers, we can use the property that LCM(a, b) = (a * b) / GCD(a, b), where GCD(a, b) is the greatest common divisor of a and b.

So, LCM(n, n/2) = (n * (n/2)) / GCD(n, n/2) = (n^2) / GCD(n, n/2).

3. Similarly, LCM(n, n/3) = (n * (n/3)) / GCD(n, n/3) = (n^2) / GCD(n, n/3).

4. We are looking for n such that n = LCM(n^2/GCD(n, n/2), n^2/GCD(n, n/3)).

5. The minimum value of n can be found by iterating through numbers starting from 1 and checking if n = LCM(n^2/GCD(n, n/2), n^2/GCD(n, n/3)).

So, you will need to iterate through values of "n" starting from 1, and compute the LCM expression until you find the minimum value of n that satisfies the equation.