Show that cube of any positive integer is of the form 4m, 4m + 1 or 4m + 3, for some integer m.
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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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Show that cube of any positive integer is of the form 4m, 4m + 1 or 4m + 3, for some integer m.
How to assume value of “b”? Here it is assumed as a=4q+0,1,2,3. But in some similar problems value of b is taken as 3. So what is the criteria to assume...
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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 cube of a positive integer of the form 6q + r, q is an integer and r = 0, 1, 2, 3, 4, 5 is also of the form 6m+r.
Solution: 6q + r is a positive integer, where q is an integer and r = 0, 1, 2, 3, 4, 5 Then, the positive integers are of the form 6q, 6q+1,...
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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 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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Show that any positive odd integer is of the form 6q + 1 or 6q + 3 or 6q + 5, where q is some integer
Let a be any positive odd integer and b = 6. Then, by Euclid’s algorithm, a = 6q + r, for some integer q ≥ 0 and 0 ≤ r...
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