Question

show that the square of positive integer cannot be from 5q+2or5q+3 for any integer q​

Answer 1

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.

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