35377

First, I started by looking for two rows or columns which had at least one letter in common. I started with ones which only had 2 letters:
a and b = LCM 45 (top row)
a and m = LCM 30 (bottom row)
The only two numbers between 1 and 9 which have an LCM of 45 are 5 and 9. Likewise, the only two numbers which have an LCM of 30 are 5 and 6. As 5 is the common number and a is the common letter, I knew that:
a = 5, and I deducted from this that b = 9 and m = 6.
I filled in these numbers and looked for rows and columns which had the letters a,b or m as well as other numbers:
a,b and e = LCM 90 (second row from the left)
I can culculate that e has to be 2, because if it was 1,3 or 9- the LCM of the three numbers whould be 45. I filled in all the e's in the sudoku grid and then looked at the second row from the right:
n,d and m = LCM 6
We know that m = 6 and e = 2; the only remaining factors of 6 are 3 and 1 so-
n/d = 3/1 i.e. I didn't yet know which letter was which number.
After this, I looked at the row which had 56 as its LCM (the very middle row) and had two letters- c and w. I knew that these letters must be 7 and 8 because they are the only numbers between 1 and 9 which have an LCM of 56. The letter c also lies in the first column on the left, which has an LCM of 28. As 7 is a multiple of 28 and 56 and 8 isn't, I could conclude that:
w = 8
c = 7 From this I could also conclude that:
v = 4 (in the first column), as the LCM of 4 and 7 is 28.
After these figures were filled in, I was left with two numbers- 1 and 3, and two letters- n and d. When I looked at the second row to the top, where 4, 8 and n had an LCM of 8, I concluded that n = 1 and d = 3 because 8 is not a multiple of 3 but is a multiple of 1.
I know that I got all the numbers right because I was able to fill in the sudoku (it is uploaded).