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If n is a positive integer, then n(n + 1)(n + 2) is
- a)even only when n is even
- b)even only when n is odd
- c)odd whenever n is odd
- d)divisible by 3 only when n is odd
- e)divisible by 4 whenever n is even
Correct answer is option 'E'. Can you explain this answer?
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If n is a positive integer, then n(n + 1)(n + 2) isa)even only when n ...
(E) divisible by 4 whenever n is even
First, let's analyze the given expression n(n - 1)(n - 2).
Notice that the three factors are consecutive integers. This means that at least one of them must be even (divisible by 2), and at least one of them must be divisible by 3.
(A) is incorrect because if n is odd, then (n - 1) is even, so the product is even.
(B) is incorrect because if n is even, then n is even, so the product is even.
(C) is incorrect because if n is odd, then (n - 1) is even, so the product is even.
(D) is incorrect because if n is even, then either n or (n - 2) is divisible by 3, so the product is divisible by 3.
(E) is correct because if n is even, then n is divisible by 2. Furthermore, since the three factors are consecutive integers, one of the other two factors (n - 1) or (n - 2) must also be divisible by 2. Therefore, the product must be divisible by 4.
First, let's analyze the given expression n(n - 1)(n - 2).
Notice that the three factors are consecutive integers. This means that at least one of them must be even (divisible by 2), and at least one of them must be divisible by 3.
(A) is incorrect because if n is odd, then (n - 1) is even, so the product is even.
(B) is incorrect because if n is even, then n is even, so the product is even.
(C) is incorrect because if n is odd, then (n - 1) is even, so the product is even.
(D) is incorrect because if n is even, then either n or (n - 2) is divisible by 3, so the product is divisible by 3.
(E) is correct because if n is even, then n is divisible by 2. Furthermore, since the three factors are consecutive integers, one of the other two factors (n - 1) or (n - 2) must also be divisible by 2. Therefore, the product must be divisible by 4.
Most Upvoted Answer
If n is a positive integer, then n(n + 1)(n + 2) isa)even only when n ...
Solution:
Given expression is n(n-1)(n-2)
Let's find out the values of the expression for different values of n.
For n=1, the expression becomes 1(0)(-1) = 0
For n=2, the expression becomes 2(1)(0) = 0
For n=3, the expression becomes 3(2)(1) = 6
For n=4, the expression becomes 4(3)(2) = 24
For n=5, the expression becomes 5(4)(3) = 60
From the above values, we can observe that:
- The expression is always even when n is even.
- The expression is never even when n is odd.
- The expression is divisible by 3 whenever n is a multiple of 3.
- The expression is divisible by 4 whenever n is even.
Therefore, the correct option is C, i.e., the expression is divisible by 4 whenever n is even.
Given expression is n(n-1)(n-2)
Let's find out the values of the expression for different values of n.
For n=1, the expression becomes 1(0)(-1) = 0
For n=2, the expression becomes 2(1)(0) = 0
For n=3, the expression becomes 3(2)(1) = 6
For n=4, the expression becomes 4(3)(2) = 24
For n=5, the expression becomes 5(4)(3) = 60
From the above values, we can observe that:
- The expression is always even when n is even.
- The expression is never even when n is odd.
- The expression is divisible by 3 whenever n is a multiple of 3.
- The expression is divisible by 4 whenever n is even.
Therefore, the correct option is C, i.e., the expression is divisible by 4 whenever n is even.
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If n is a positive integer, then n(n + 1)(n + 2) isa)even only when n is evenb)even only when n is oddc)odd whenever n is oddd)divisible by 3 only when n is odde)divisible by 4 whenever n is evenCorrect answer is option 'E'. Can you explain this answer?
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