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A positive integer is of the form 3q + 1, q being a natural number. Can you write its square in any form other than 3m + 1, i.e., 3m or 3m + 2 for some integer m? Justify your answer. |
Answer» No. Justification: Consider the positive integer 3q + 1, where q is a natural number. (3q + 1)2 = 9q2 + 6q + 1 = 3(3q2 + 2q) + 1 = 3m + 1, (where m is an integer which is equal to 3q2 + 2q. Thus (3q + 1)2 cannot be expressed in any other form apart from 3m + 1. |
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