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Which of the following must be true for three positive consecutive integers?
I. Their product is always divisible by 6.
II. Their sum is always divisible by 3.
III. Their product is always divisible by 4.
II. Their sum is always divisible by 3.
III. Their product is always divisible by 4.
- a)I only
- b)II only
- c)II and III
- d)I and II
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Which of the following must be true for three positive consecutive int...
Let us consider two sets of positive consecutive integers starting with an odd number and an even number, respectively i.e. (1, 2, 3) and (2, 3, 4).
I is true in both cases.
II is true in both cases.
III is not true in the first case.
So, (4) is the answer.
I is true in both cases.
II is true in both cases.
III is not true in the first case.
So, (4) is the answer.
Most Upvoted Answer
Which of the following must be true for three positive consecutive int...
To determine which statements are true for three positive consecutive integers, let's consider the integers as n, n+1, and n+2.
I. Their product is always divisible by 6.
To check if the product of the three integers is divisible by 6, we need to see if at least one of the integers is divisible by 6.
If n is divisible by 6, then n(n+1)(n+2) will be divisible by 6.
If n+1 is divisible by 6, then (n+1)(n+2)n will be divisible by 6.
If n+2 is divisible by 6, then (n+2)n(n+1) will be divisible by 6.
Therefore, in any case, the product of the three integers will be divisible by 6. Hence, statement I is true.
II. Their sum is always divisible by 3.
The sum of the three integers can be represented as n + (n+1) + (n+2) = 3n + 3.
3n + 3 is divisible by 3 for any value of n. Therefore, the sum of the three integers is always divisible by 3. Hence, statement II is true.
III. Their product is always divisible by 4.
To check if the product of the three integers is divisible by 4, we need to see if at least two of the integers are even.
If n and n+1 are even, then n(n+1)(n+2) will be divisible by 4.
If n+1 and n+2 are even, then (n+1)(n+2)n will be divisible by 4.
If n and n+2 are even, then (n+2)n(n+1) will be divisible by 4.
However, it is not guaranteed that at least two of the three consecutive integers will be even. For example, the consecutive integers 1, 2, 3 do not satisfy this condition. Therefore, statement III is not always true.
Therefore, the correct answer is option 'D' - I and II.
I. Their product is always divisible by 6.
To check if the product of the three integers is divisible by 6, we need to see if at least one of the integers is divisible by 6.
If n is divisible by 6, then n(n+1)(n+2) will be divisible by 6.
If n+1 is divisible by 6, then (n+1)(n+2)n will be divisible by 6.
If n+2 is divisible by 6, then (n+2)n(n+1) will be divisible by 6.
Therefore, in any case, the product of the three integers will be divisible by 6. Hence, statement I is true.
II. Their sum is always divisible by 3.
The sum of the three integers can be represented as n + (n+1) + (n+2) = 3n + 3.
3n + 3 is divisible by 3 for any value of n. Therefore, the sum of the three integers is always divisible by 3. Hence, statement II is true.
III. Their product is always divisible by 4.
To check if the product of the three integers is divisible by 4, we need to see if at least two of the integers are even.
If n and n+1 are even, then n(n+1)(n+2) will be divisible by 4.
If n+1 and n+2 are even, then (n+1)(n+2)n will be divisible by 4.
If n and n+2 are even, then (n+2)n(n+1) will be divisible by 4.
However, it is not guaranteed that at least two of the three consecutive integers will be even. For example, the consecutive integers 1, 2, 3 do not satisfy this condition. Therefore, statement III is not always true.
Therefore, the correct answer is option 'D' - I and II.
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Which of the following must be true for three positive consecutive integers?I. Their product is always divisible by 6.II. Their sum is always divisible by 3.III. Their product is always divisible by 4.a)I onlyb)II onlyc)II and IIId)I and IICorrect answer is option 'D'. Can you explain this answer?
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