It is often easier to disprove a conjecture than to try and prove it to be true. You have but to find the exception to the conjecture. We received only one contribution disproving the conjecture and that was the work of Tim from Gravesend Grammar School for Boys in Gravesend, Kent. His work is quite detailed and is as follows:

A computer search showed that this is not true: the first example is 1002, which is 167*6, has eight factors (1, 2, 3, 6, 167, 334, 501, 1002) and 1001 also has eight factors (1, 7, 11, 13, 77, 91, 143, 1001). The next example is 1086 and 1085, and after that 1266 and 1275. The first example where an adjacent number has more factors than the multiple of six is 2274, which has 8 factors, and 2275, which has 12 factors. The number 6546 has 8 factors, while 6545 has 16.