For any positive integer a and 3, there exist unique integers q and r such that a = 3q + r, where r must satisfy
(A) 0 ≤ r < 3 (B) 1 < r < 3
(C) 0 < r < 3 (D) 0 < r ≤ 3
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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 cube of any positive integer is of the form 4m, 4m + 1 or 4m + 3, for some integer m.
1 Answers
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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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Prove that one of any three consecutive positive integers must be divisible by 3.
Solution: Let the three consecutive positive integers be n, n + 1 and n + 2, where n is any integer. By Euclid’s division lemma, we have a = bq + r; 0 ≤...
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Euclid’s division lemma states that for two positive integers a and b, there exist unique integers q and r such that a = bq + r, where r must satisfy
Correct answer is option (C) 0 ≤ r < b
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n^2 - 1 is divisible by 8, if n is (A) an integer (B) a natural number (C) an odd integer (D) an even integer
n2 - 1 is divisible by 8, if n is an odd integer
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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...
1 Answers 1 views