Here's a method that takes advantage of the values of these particular numbers.
... 76 = 80 - 4
so
... 76 × 327 = (80 -4)×327
... = 80×327 - 4×327
Repeated doubling will give us values that are 2, 4, and 8 times 327.
... 2×327 = 327+327 = 654
... 2×654 = 4×327 = 654+654 = 1308 . . . . we'll use this later
... 2×1308 = 8×327 = 2616
We want 80×327, so we can add a zero to the end of this last:
... 80×327 = 26160
Now, we can subtract 4×327 to get 76×327
... 80×327 - 4×327 = 26160 -1308 = 24852 = 76×327
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More conventionally, you would multiply every digit of one number by every digit of the other and add the products according to their respective place values.
327 × 076 = (3×0)×10000 + (3×7 +2×0)×1000 +(3×6 +7×0 +2×7)×100 +(2×6 +7×7)×10 +(7×6)×1
... = 0 +21,000 +3,200 + 610 +42
... = 24,852
Note the pattern of partial products here. This is a method taught to/by practitioners of Vedic mathematics, and can be done in your head. At most, you would write down the partial product sums 21, 32, 61, and 42 to keep from having to carry more than one number in your head at a time.
1, there is no solution