Tom thought of a natural number, multiplied all its digits and after that he multiplied the result by the initial number. Is it possible to get 1716 as a result?

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.