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The answer is 840 = 23 x 31 x 51 71. Any factor will of the form 2a3b5c7d where a is 0,1,2 or 3, and b,c,d are each either 0 or 1. This gives 4 x 2 x 2 x 2 = 32 factors.
Note: 21 x 31 x 51 x 71 x 111 will also have 32 factors but this product is 2310 which is too large.
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