Question

How many positive three-digit integers with each digit greater than 4 are divisible by 6?

Answer 1

6 = 2 * 3
x y z
all must be divisible by 2 (therefore even) and all bigger than 4
so each is 6 or 8
eg
6 6 6 yes
6 6 8 NO
6 8 6 NO
6 8 8 NO
8 6 6 NO
8 6 8 NO
8 8 6 NO
8 8 8 YES

Answer 2

So whats the solution

Answer 3

You did it all wrong, Damon. Only the last digit has to be even (to be divisible by 2).
6 = 2*3

Last digit has two choices: 6 and 8.

For it to be divisible by 3, sum of all three digits must be a multiple of 3.
There are only so many choices, since the digits are limited to 5, 6, 7, 8, 9.
The largest possible sum is 9+9+8 = 26, and the greatest MULTIPLE OF 3 below 26 is 24.
We find the minimum this time, and its 16, and the smallest MULTIPLE OF 3 larger than 16 is 18.

Cases:
18: 556
21: 696, 786, 876, 966, 588, 678, 768, 858
24: 998

Quite a simple casework problem, and count for yourself.
Final answer: 10

Answer 4

Hey, just count the options, the answer is 16.
there's 558, 576, 588, 666, 678, 696, 756, 768, 786, 798, 858, 876, 888, 966, 978, and 996.

make sure to check your answer before calling it simple ;)