Show that the square of any positive integer cannot be of the form Sq+

1.	Show that the square of any positive integer cannot be of the form Sq+25q + 3 for any integer q
Answer» 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

Show that the square of any positive integer cannot be of the form Sq+25q + 3 for any integer q

Answer»

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