Is n ! the largest factor of x = a ( a + 1 ) ( a + 2 ) × ⋯ × ( a + n ) ?
Consider x = a ( a + 1 ) ( a + 2 ) × ⋯ × ( a + n ) .
True or False?
The largest known factor of x is n ! for a > 0 .
Clarifications: The values of x and a are unknown. The value of n is known. (In other words: Find the largest expression only including n and constants, that must be a factor of x, no matter which value a (and x ) has.)
Notation: ! denotes the factorial notation. For example: 8 ! = 1 × 2 × 3 × ⋯ × 8 .
Hint: Substitute low values for a and n to develop an intuition.
If you get this problem wrong, please see the solutions. Recommended Problems:
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1 solution
Shouldn't the largest factor is " a ( a + 1 ) ( a + 2 ) × ⋯ × ( a + n ) " itself?
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Sorry misunderstood you.
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The largest factor of any integer N is ∣ N ∣ itself.
E.g: All the positive factors of 12 are { 1 , 2 , 3 , 4 , 6 , 1 2 } , so the largest factor of 12 is 12 itself.
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@Pi Han Goh – Assumptions: The value of x and a are unknown. The value of n is known.
@Pi Han Goh – True or False? The largest known factor of x is n ! for a > 0 .
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@Ron Lauterbach – Yes, I read that. I still don't think that is clear.
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@Pi Han Goh – Does this change clarify everything?
Clarifications: The values of x and a are unknown. The value of n is known. In the question known refers to knowing an exact value and not an expression with unknown variables.
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@Ron Lauterbach – Okay, I think this is the best possible way to phrase this:
0 ! ( n + 1 ) ! , 1 ! ( n + 2 ) ! , 2 ! ( n + 3 ) ! , …
In terms of n , what is the greatest common divisor of all the numbers above?
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@Pi Han Goh – This is kind of unrelated to my problem.
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@Ron Lauterbach – First of all my problem goes from 1 to n and not from n+1 to infinity.
@Pi Han Goh – For now, I made these changes:
Clarifications: The values of x and a are unknown. The value of n is known. (In other words: Find the largest expression only including n and constants, that must be a factor of x, no matter which value a (and x ) has.)
Would you say, this is clear enough without having to rephrase the entire problem?
It says the largest known factor. I specified, that we don't know a (and even x to clarify).
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Ah I see. Then I recommend that you change your question to:
Let x a = a ( a + 1 ) ( a + 2 ) ⋯ ( a + n ) for positive integers a and n .
True or False?
The greatest common divisor of x a , x a + 1 , x a + 2 , … is n ! .
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@Pi Han Goh – The answer to the question is still valid, even if the question is misunderstood, isn't it?
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@Ron Lauterbach – Yes, this is a good question, but it's just hard to understand what you meant.
@Pi Han Goh – How do you create these boxes?
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@Pi Han Goh – In your example it is unclear what x a , x a + 1 , x a + 2 , … refers to.
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@Ron Lauterbach – I've already written the definition of x a .
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@Pi Han Goh – Yes, but writing x a ( x a + 1 , ... is confusing (in general).
I didn't change anything so far. What I said previously (now deleted) was referring to a typo (It said + not × .
The largest known factor is ( n + 1 ) ! , as there are n + 1 terms (don't forget a ).
( n + 1 ) ! > n !
The correct answer is F a l s e .