Prove that 6nn 1n 2 for any integer n 1SolutionWe will fir

Prove that 6|n(n + 1)(n + 2) for any integer n 1.

Solution

We will first show that the product of any three consecutive integers ( each 1) is a multiple of 3. Since any integer can be written in the form 3n -1, 3n or 3n + 1, the difference of two integers of the same form is a multiple of 3 and therefore, not less than 3. But, the difference of any 2 of the consecutive integers is less than 3. Thus, three consecutive integers are respectively of the above three forms viz. 3n -1, 3n or 3n + 1, among which one is of the form 3n which is a multiple of 3.Thus,the product of any three consecutive integers ( each 1) is a multiple of 3.

Since one of the any two consecutive integers ( each 1) has to be even, and thus divisible by 2, the product of product of any three consecutive integers ( each 1) also is a multiple of 2. Since the product of any three consecutive integers ( each 1) is a multiple of 3.also (proved above), the product of any three consecutive integers ( each 1) is a multiple of 6. Therefore, 6 divides n(n+1)(n +2) for any integer n 1..