Question

Prove that every natural number can be written in the form 5k or 5k + 1 or 5k + 2, k ∈ N ∪ {0}

Answer 1

question is -----> Prove that every natural number can be written in the form 5k or 5k ± 1 or 5k ± 2, k ∈ N ∪ {0}

solution :- according to Euclid algorithm, every natural , a can be written as a = bk + m
where 0 ≤ m < b
Let b = 5 , then a = 5k + m
and then, m =0, 1 , 2 , 3 , 4
so, all natural number of a can be written as 5k, 5k + 1 , 5k +2 , 5k +3 and 5k + 4 where k ∈ N ∪ {0}.

a = 5k + 3
= 5k + 5 - 2
= 5(k + 1) - 2 = 5k' -2 , k' = k +1 where k ∈ N ∪ {0}.
= 5k - 2, [ taking k intead of k' ]
hence, 5k +3 can be written as 5k -2

now, a = 5k + 4
= 5k + 5 - 1
= 5(k + 1) - 1 = 5k' - 1 , k' = k + 1 where k ∈ N ∪ {0}.
= 5k - 1 [ taking k instead of k']
hence, 5k +4 can be written as 5k -1

hence, it is clear that all natural number can be written as 5k , 5k ±1 and 5k ±2 for all k∈ N ∪ {0}.