All of these operations result in natural numbers, so let's consider all divisor pairs of 1716:
(1, 1716)
(2, 858)
(3, 572)
(4, 429)
(6, 286)
(11, 156)
(12, 143)
(13, 132)
(22, 78)
(26, 66)
(33, 52)
(39, 44)
The divisor pair (12, 143) fits the bill, since the product of the digits of 143 is 1*4*3 = 12, so the answer is yes.
