Show that the square of any positive integer cannot be of the form 5q + 2 or 5q + 3 for any integer q
Solution:
Assume the positive integer to be = a
Using Euclid’s division lemma,
a = bm + r.
It is given that,
b = 5.
a = 5m + r.
So, r = 0, 1, 2, 3, 4.
Case 1: When r = 0, a = 5m.
Case 2: When r = 1, a = 5m + 1.
Case 3: When r = 2, a = 5m + 2.
Case 4: When r = 3, a = 5m + 3.
Case 5: When r = 4, a = 5m + 4.
Case 1: When a = 5m
a2 = (5m)2 = 25m2
a2 = 5(5m2) = 5q, where q = 5m2.
Case 2: When a = 5m + 1
a2 = (5m2 + 1)2 = 25m2 + 10 m + 1
a2 = 5(5m2 + 2m) + 1 = 5q + 1, where q = 5m2 + 2m.
Case 3: When a = 5m + 2
a2 = (5m + 2)2
a2 = 25m2 + 20m + 4
a2 = 5(5m2 + 4m) + 4
a2 = 5q + 4 where q = 5m2 + 4m.
Case 4:When a = 5m + 3
a2 = (5m + 3)2 = 25m2 + 30m + 9
a2 = 5(5m2 + 6m + 1) + 4
a2 = 5q + 4 where q = 5m2 + 6m + 1.
Case 5: When a = 5m + 4
a2 = (5m + 4)2 = 25m2 + 40m + 16
a2 = 5(5m2 + 8m + 3) + 1
a2 = 5q + 1 where q = 5m2 + 8m + 3
Therefore, the square of any positive integer cannot be of the form 5q + 2 or 5q + 3
✦ Try This: Show that the square of any positive integer cannot be of the form (5m + 2) or (5m + 5) for any integer m
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 1
NCERT Exemplar Class 10 Maths Exercise 1.3 Problem 3
Show that the square of any positive integer cannot be of the form 5q + 2 or 5q + 3 for any integer q
Summary:
The square of any positive integer cannot be of the form 5q + 2 or 5q + 3 for any integer q
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