The naive way would be to linearly iterate the range and multiply with each number in the range.
Example: Array: {1,2,3,4,5,6,7,8,9,10}; Multiply index 3 to index 8 wit
By BIT you can also update a range.(Range update by BIT).
You can also use Segment Tree, RMQ. You can multiply with a given number easily.
Nice tutorial about ST and RMQ. RMQ Topcoder and Segment Tree