How to count no of rows in a data frame whose values divisible by 3 or 5?

Compare modulo 5 and 3 with 0 and filter by boolean indexing with | for bitwise OR:

df = df[(df['ones'] % 5 == 0) | (df['zeros'] % 3 == 0)]
print (df)
      ones  zeros
5       10      6
6       11      6
9        4      6
13      14      3
26       6     12
29       9      9
4339    10      9
4342     4      6
4344     5     11
4345     7      9
4351     5      8
4352     5      7
4353     5      8
4355    11      3
4357     7      6
4358     8     12
4362     8      3
4364     9      3
4366     9      6
4368     4      3

If need to count number of matched values:

out = ((df['ones'] % 5 == 0) | (df['zeros'] % 3 == 0)).sum()
print (out)
20

We can also use the query method to filter, combined with the pandas methods Series.mod and eq which stands for equal:

df = df.query('ones.mod(5).eq(0) | zeros.mod(3).eq(0)')

    ones  zeros
5     10      6
6     11      6
9      4      6
13    14      3
26     6     12
29     9      9
30    10      9
33     4      6
35     5     11
36     7      9
42     5      8
43     5      7
44     5      8
46    11      3
48     7      6
49     8     12
53     8      3
55     9      3
57     9      6
59     4      3