Question

Show that the square of any positive integer cannot be of the form 5q+2 or 5q+3 for any integer q.

Answer 1

Sol: Consider a be any positive integer. By Euclid's division lemma, a = bn + r  where  b = 5 ⇒ a = 5n + r So that r can be any of 0, 1, 2, 3, 4 ∴  a = 5n  when  r = 0  a = 5n + 1  when  r = 1  a = 5n + 2  when  r = 2  a = 5n + 3  when  r = 3  a = 5n + 4  when  r = 4So, "a"  is any positive integer in the form of 5n,  5n + 1 ,  5n + 2 , 5n + 3 ,  5n + 4 for some integer n. Case i :  a = 5n  ⇒  a 2 = (5n)2 = 25n 2  ⇒  a 2 = 5(5n 2)          = 5q , where  q = 5n 2 Case ii :  a = 5n + 1    ⇒  a 2 = (5n + 1)2 = 25n 2 + 10 n + 1    ⇒  a 2 = 5 (5n 2 + 2n) + 1            = 5q + 1,  where q = 5n 2 + 2n . Case iii :   a = 5n + 2  ⇒   a 2 = (5n + 2)2  =  25n 2 + 20n +4  =  25n 2 + 20n +4  =  5 (5n 2 + 4n) + 4  =  5q + 4  where q = 5n 2 + 4n   Case iv:  a = 5n + 3  ⇒  a 2= (5n + 3)2 = 25n 2 + 30n + 9  = 25n 2 + 30n + 5 + 4  = 5 (5n 2 + 6n + 1) + 4  = 5q + 4  where  q = 5n 2 + 6n + 1   Case v:    a = 5n + 4  ⇒  a 2 = (5n + 4)2 = 25n 2 + 40n + 16  = 25n 2 + 40n + 15 + 1  = 5 (5n 2 + 8n + 3) + 1  = 5q + 1  where  q = 5n 2 + 8n + 3From all these cases, it is clear that square of any positive integer can not be of the form 5q + 2  or  5q + 3 for any integer q.

Answer 2

Answer:

Step-by-step explanation:

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, 3, 4 because 0 ≤ r < 5. So, a = 5m or 5m + 1 or 5m + 2 or 5m + 3 or 5m + 4. So, (5m)2 = 25m2 = 5(5m2) = 5q, where q is any integer. (5m + 1)2 = 25m2 + 10m + 1 = 5(5m2 + 2m) + 1 = 5q + 1, where q is any integer. (5m + 2)2 = 25m2 + 20m + 4 = 5(5m2 + 4m) + 4 = 5q + 4, where q is any integer. (5m + 3)2 = 25m2 + 30m + 9 = 5(5m2 + 6m + 1) + 4 = 5q + 4, where q is any integer. (5m + 4)2 = 25m2 + 40m + 16 = 5(5m2 + 8m + 3) + 1 = 5q + 1, where q is any integer. Hence, The square of any positive integer is of the form 5q, 5q + 1, 5q + 4 and cannot be of the form 5q + 2 or 5q + 3 for any integer q.Read more on Sarthaks.com - https://www.sarthaks.com/12813/show-that-the-square-of-any-positive-integer-cannot-be-of-the-form-5q-2-or-5q-3-for-any-integer