Prove that the square of any positive integer is of the form 3m or, 3m + 1 but not of the form 3m +2.
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Show that any positive odd integer is of the form 6q+1,6q+3 & 6q+5 where q is some integer.
Solution: Let a be any positive integer and b = 6. Then, by Euclid’s algorithm, a = 6q + r for some integer q ≥ 0, and r = 0, 1, 2, 3,...
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Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.
Solution: Let a be any positive integer and b = 6. Then, by Euclid’s algorithm, a = 6q + r for some integer q ≥ 0 and r = 0,1,2,3,4,5 because 0 ≤ r ≤ 6. So, a...
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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
Solution: 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,...
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Show that the square of any positive integer cannot be of the form 5q + 2 or 5q + 3 for any integer q.
Solution: 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,...
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Show that the square of any positive integer cannot be of the form 6m + 2 or 6m + 5 for any integer m.
Solution: Let a be the positive integer and b = 6. Then, by Euclid’s algorithm, a = 6q + r for some integer q ≥ 0 and r = 0, 1, 2,...
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Show that the square of any odd integer is of the form 4q + 1, for some integer q.
Solution: Let a be any odd integer and b = 4. Then, by Euclid’s algorithm, a = 4m + r for some integer m ≥ 0 and r = 0,1,2,3 because 0 ≤ r <...
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Show that the square of an odd positive integer is of the form 8q + 1, for some integer q.
By Euclid’s division algorithm a = bq + r, where 0 ≤ r ≤ b Put b = 4 a = 4q + r, where 0 ≤ r ≤ 4 If r = 0,...
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Prove that the square of any positive integer is of the form 3m or, 3m + 1 but not of the form 3m +2.
VERY NICE EXPLANATION
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Prove that if a positive integer is of the form 6q + 5, then it is of the form 3q + 2 for some integer q, but not conversely.
Let, n = 6q + 5, when q is a positive integer We know that any positive integer is of the form 3k, or 3k + 1, or 3k + 2 ∴...
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Prove that the square of any positive integer of the form 5q + 1 is of the same form.
Let n = 5q + 1 where q is a positive integer ∴ n2 = (5q + 1)2 = 25q2 + 10q + 1 = 5(5q2 + 2q) + 1 = 5m + 1,...
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