There seems to be only 11 powers of ten that are products of 2 integers without any zero digits.
10 ^ 0 = 1 * 1
10 ^ 1 = 2 * 5
10 ^ 2 = 4 * 25
10 ^ 3 = 8 * 125
10 ^ 4 = 6 * 625
10 ^ 5 = 32 * 3125
10 ^ 6 = 64 * 15625
10 ^ 7 = 128 * 78125
10 ^ 9 = 512 * 1953125
10 ^ 18 = 262144 * 3814697265625
10 ^ 33 = 8589934592 * 116415321826934814453125
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