Use digital roots.
Digital root of an integer is, by definition, the sum of its digits computed recursively until only one digit remains. Thus to compute the digital root of 33572, we calculate first 3 + 3 + 5 + 7 + 2 = 20. The digitl root of 33572 is then 2 = 2 + 0.
The important fact about digital roots is that the digital root of the product of integers equals the product of their digital roots (downsized to a digital root if necessary.)
348·11 = 3828. The digital roots of 348, 11, and 3828 are respectively 6, 2, and 3. Take the product of 6 and 2: 6·2 = 12 with the digital root equal to 1 + 2 = 3, which is exactly the digital root of 3828.
(Thanks are due to Jayesh Patel from Gujarat State (India) for reminding me of this useful property of the digital roots.)