To be a multiple of 2, the number must end with 4, 6, or 8

So we choose can the 5th digit any one of 3 ways to make it even.
We can then choose the first digit any of the remaining 4 digits.
We can then choose the second digit any of the remaining 3 digits.
We can then choose the third digit either of the remaining 2 digits.
We then must choose the fourth digit as only the remaining 1 digit.

Answer = (3)(4)(3)(2)(1) = 72

Here they all are, 9 rows of 8 numbers each.

45678, 45768, 45786, 45876, 46578, 46758, 47568, 47586, 
47658, 47856, 48576, 48756, 54678, 54768, 54786, 54876, 
56478, 56748, 56784, 56874, 57468, 57486, 57648, 57684, 
57846, 57864, 58476, 58674, 58746, 58764, 64578, 64758, 
65478, 65748, 65784, 65874, 67458, 67548, 67584, 67854, 
68574, 68754, 74568, 74586, 74658, 74856, 75468, 75486, 
75648, 75684, 75846, 75864, 76458, 76548, 76584, 76854, 
78456, 78546, 78564, 78654, 84576, 84756, 85476, 85674, 
85746, 85764, 86574, 86754, 87456, 87546, 87564, 87654, 



To be a multiple of 5, the number must end with 5

We must choose the fifth digit only 1 way, as 5, to make it be a multiple of 5.
We can then choose the first digit any of the remaining 4 digits.
We can then choose the second digit any of the remaining 3 digits.
We can then choose the third digit either of the remaining 2 digits.
We then must choose the fourth digit as only the remaining 1 digit.

Answer = (1)(4)(3)(2)(1) = 24

Here they all are, 3 rows of 8 numbers each.

46785, 46875, 47685, 47865, 48675, 48765, 64785, 64875, 
67485, 67845, 68475, 68745, 74685, 74865, 76485, 76845, 
78465, 78645, 84675, 84765, 86475, 86745, 87465, 87645.

Edwin