21978 when reversed gives 87912. So the units digit of the product must be 2. 

In the original number, the unit digit is 8. To get 2 in the unit digit in the product, 8 must be multiplied with either 4 or 9.

$8\times 4=32$ or $8\times 9=72$

The tens digit in the reversed number 87912 is 1. To get that, we need to multiply by 4.

$2\stackrel{3}{1}\stackrel{3}{9}\stackrel{3}{7}8\times \underset{–}{4}\underset{–}{87912}$

Hence,  21978 is multiplied by '4' to get a product where the digits are exactly reversed.