A positive integer is of the form 3q + 1, q being a natural number. Can you write its square in any form other than 3m + 1

A = {x: x is a natural number} = {1, 2, 3, 4, 5 …}  B ={x: x is an even natural number} = {2, 4, 6, 8 …}  C = {x:...

Solution: No, every positive integer cannot be only of the form 4q + 2. Justification: Let a be any positive integer. Then by Euclid’s division lemma, we have a = bq + r, where...

Solution: No. Justification: Let a be any positive integer. Then by Euclid’s division lemma, we have a = bq + r, where 0 ≤ r < b For b = 3, we have a = 3q...

Solution: Let a be the positive integer and b = 5. Then, by Euclid’s algorithm, a = 5m + r for some integer m ≥ 0 and r = 0, 1, 2,...

Solution: Let a be the positive integer and b = 6. Then, by Euclid’s algorithm, a = 6q + r for some integer q ≥ 0 and r = 0, 1, 2,...

n2 - 1 is divisible by 8, if n is an odd integer 

By Euclid’s division algorithm a = bq + r, where 0 ≤ r ≤ b Put b = 4 a = 4q + r, where 0 ≤ r ≤ 4 If r = 0,...

VERY NICE EXPLANATION

Let n = 5q + 1 where q is a positive integer ∴ n2 = (5q + 1)2 = 25q2 + 10q + 1 = 5(5q2 + 2q) + 1 = 5m + 1,...