YES.
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Yes, the product of any three consecutive numbers is divisible by 6. This is because among any three consecutive integers, at least one of them is even (ensuring divisibility by 2), and at least one of them is divisible by 3. Since 6 is the product of 2 and 3, the product of any three consecutive numbers is therefore divisible by 6.
420
There are only two prime numbers that are consecutive numbers, 2 and 3. Their product is 2 x 3 = 6. The first prime numbers are 2, 3, 5, and 7 and the only two consecutive prime numbers whose product is a single digit are 2 and 3. (The next two consecutive prime numbers, 3 and 5, have a two-digit product.)
There is no set of 3 consecutive odd numbers whose product is 6,873. However, there is a set of 3 consecutive odd numbers whose sum is 6,873: 2289, 2291, and 2293.
There are no such numbers. The smallest set of three consecutive composite numbers is {8, 9, 10} and the product of these numbers is 720.