For any positive integer n, prove that n3 – n is divisible by 6.
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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 and only one out of n, n + 2 and n + 4 is divisible by 3, where n is any positive integer.
Solution: By Euclid’s division lemma, we have a = bq + r; 0 ≤ r < b For a = n and b = 3, we have n = 3q + r, …(i) where q is...
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For any positive integer n, prove that n^3 – n is divisible by 6.
Let a = n^3 - n ⇒ a = n.(n^2 - 1) ⇒ a = n.(n-1).(n+1) [∵(a^2 - b^2) = (a-b)(a+b)] ⇒ a = (n-1).n.(n+1) ...(i) We know that, 1. If a number is...
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Show that one and only one out of n, n + 4, n + 8, n + 12 and n + 16 is divisible by 5, where n is any positive integer.
Solution: By Euclid’s division lemma, we have a = bq + r; 0 ≤ r < b For a = n and b = 5, we have n = 5q + r …(i) Where q is...
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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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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 product of three consecutive positive integer is divisible by 6.
Let, n be any positive integer. Since any positive integer is of the form 6q or 6q + 1 or 6q + 2 or, 6q + 3 or 6q +...
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For any positive integer n , prove that n^3 − n divisible by 6.
Another solution would be: n^3 - n = n (n^2 - 1) = n (n - 1) (n + 1) Whenever a number is divided by 3, the remainder obtained is...
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For any positive integer n , prove that n^3 − n divisible by 6.
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