Question

State fundamental theorem of arithmetic. Prove that only one out of n, n+2 and n+4
is divisible by 3​

Answer 1

In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors.For example,

1200 = 24 × 31 × 52 = 2 × 2 × 2 × 2 × 3 × 5 × 5 = 5 × 2 × 5 × 2 × 3 × 2 × 2

One out of n,n+2 and n+4 is divisible because there is a gap of 2 and 4

If n is divisible by 3 then other 2 will not

If the gap was of 3 then if n would be divisible by 3 then other 2 will also and vice versa.

Coming back to your problem,

Even if n is not divisible but n+2 or n+4 will be because if n is a(the largest multipleof 3 before n) -2 or -4 , 2 or 4 will be added and will make it a multiple of 3. Thank you