Question

Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump.

Cozy goes two steps up with each jump (though if necessary, he will just jump the last step).

Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than $5$ steps left).

Suppose Dash takes $19$ fewer jumps than Cozy to reach the top of the staircase. Let $s$ denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of $s?$

$9$

$11$

$12$

$13$

$15$

Solution:

Suppose Dash takes $d+1$ jumps. Then the number of steps $t$ is one of $5d+1,5d+2,5d+3,5d+4,5d+5.$ Cozy takes $19$ more jumps, so Cozy takes $d+20$ jumps, which means $t$ is either $2d+39$ or $2d+40$ .

Matching these possibilities, the integer solutions are $5d+3=2d+39,\quad 5d+1=2d+40,\quad 5d+4=2d+40.$ They give $t=63,66,64$ , respectively.

Thus $s=63+66+64=193$ , and the sum of its digits is $13$ .

Thus, the correct answer is D.

2015 AMC 10B Problem 21

Solution:

Problem 21 in Other Years