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The function f(m) is defined for all positive integers m as the product of m + 4, m + 5, and m + 6. If n is a positive integer, then f(n) must be divisible by which one of the following numbers?
- a)4
- b)5
- c)6
- d)7
- e)11
Correct answer is option 'C'. Can you explain this answer?
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The function f(m) is defined for all positive integers m as the produc...
Understanding the Function f(m)
The function \( f(m) \) is defined as:
\[ f(m) = (m + 4)(m + 5)(m + 6) \]
This function represents the product of three consecutive integers starting from \( m + 4 \).
Divisibility Analysis
To determine the divisibility of \( f(n) \) by various numbers, we can analyze the product of three consecutive integers:
- Consecutive Integers: Among any three consecutive integers, at least one integer is divisible by 2, and at least one is divisible by 3.
Divisibility by 4
- Checking divisibility by 4: In three consecutive integers, at least one of them is even. However, it does not guarantee that one of them is divisible by 4.
Divisibility by 5
- Checking divisibility by 5: Among three consecutive integers, there may or may not be a number divisible by 5.
Divisibility by 6
- Checking divisibility by 6: Since one of the integers is guaranteed to be even (ensuring a factor of 2) and one must be divisible by 3, the product \( f(n) \) is guaranteed to be divisible by \( 2 \times 3 = 6 \).
Divisibility by 7
- Checking divisibility by 7: Among three consecutive integers, there is no guarantee that one of them is divisible by 7.
Divisibility by 11
- Checking divisibility by 11: Similarly, among three consecutive integers, there is no guarantee for divisibility by 11.
Conclusion
Thus, the only number among the options that \( f(n) \) is guaranteed to be divisible by is:
- Answer: c) 6
This confirms that \( f(n) \) must be divisible by 6 based on the properties of consecutive integers.
The function \( f(m) \) is defined as:
\[ f(m) = (m + 4)(m + 5)(m + 6) \]
This function represents the product of three consecutive integers starting from \( m + 4 \).
Divisibility Analysis
To determine the divisibility of \( f(n) \) by various numbers, we can analyze the product of three consecutive integers:
- Consecutive Integers: Among any three consecutive integers, at least one integer is divisible by 2, and at least one is divisible by 3.
Divisibility by 4
- Checking divisibility by 4: In three consecutive integers, at least one of them is even. However, it does not guarantee that one of them is divisible by 4.
Divisibility by 5
- Checking divisibility by 5: Among three consecutive integers, there may or may not be a number divisible by 5.
Divisibility by 6
- Checking divisibility by 6: Since one of the integers is guaranteed to be even (ensuring a factor of 2) and one must be divisible by 3, the product \( f(n) \) is guaranteed to be divisible by \( 2 \times 3 = 6 \).
Divisibility by 7
- Checking divisibility by 7: Among three consecutive integers, there is no guarantee that one of them is divisible by 7.
Divisibility by 11
- Checking divisibility by 11: Similarly, among three consecutive integers, there is no guarantee for divisibility by 11.
Conclusion
Thus, the only number among the options that \( f(n) \) is guaranteed to be divisible by is:
- Answer: c) 6
This confirms that \( f(n) \) must be divisible by 6 based on the properties of consecutive integers.
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The function f(m) is defined for all positive integers m as the product of m + 4, m + 5, and m + 6. If n is a positive integer, then f(n) must be divisible by which one of the following numbers?a)4b)5c)6d)7e)11Correct answer is option 'C'. Can you explain this answer?
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