Question

show that the square of an odd positive integer can be of the form 6q + 1 or 6q + 3 for some integer q​

Answer 1

Answer:

Step-by-step explanation:

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Answer 2

ANSWER

We know that any positive integer can be of the form 6m, 6m + 1, 6m + 2,

6m + 3, 6m + 4 or 6m + 5, for some integer m.

Thus, an odd positive integer can be of the form 6m + 1, 6m + 3, or 6m + 5

Thus we have:

(6 m +1)2 = 36 m2 + 12 m + 1 = 6 (6 m2 + 2 m) + 1 = 6 q + 1, q is an integer

(6 m + 3)2 = 36 m2 + 36 m + 9 = 6 (6 m2 + 6 m + 1) + 3 = 6 q + 3, q is an integer

(6 m + 5)2 = 36 m2 + 60 m + 25 = 6 (6 m2 + 10 m + 4) + 1 = 6 q + 1, q is an integer.

Thus, the square of an odd positive integer can be of the form 6q + 1 or 6q + 3.

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