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Thread: The "Length" of n

  1. #1
    Senior Member harpazo's Avatar
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    The "Length" of n

    NOTE: CAN SOMEONE MOVE THIS QUESTION TO THE ALGEBRA SECTION? I posted here by mistake.

    For any positive integer n, n > 1, the LENGTH of n is the number of positive primes (not necessarily distinct) whose product is n. For example, the length of 50 is 3 since 50 = (2)(5)(5).

    Which of the following integers has length 3?

    A. 3
    B. 15
    C. 60
    D. 64
    E. 105

    I selected 60 but the correct answer is 105.

    My Work:

    60 = 15 * 2 * 2

    I forgot that 15 can also be broken down a little more via prime factorization or the famous factor tree. I understand why my answer is wrong. Also, 15 is not a prime number. The question (like most recent posted applications) comes from the GMAT Review/Prep book (1997). BTW, I am not taking the GMAT. However, I find the given word problems interestingly enough to daily practice.

    A. How would you tackle this problem?

    B. What indication is there in the question itself pointing to the idea of prime factorization?
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    MHF Contributor MarkFL's Avatar
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    Re: The "Length" of n

    The given definition of the "length" of n indicates that prime factors are to be used.
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    Senior Member harpazo's Avatar
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    Re: The "Length" of n

    Thanks. Give me another example to practice.
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    Re: The "Length" of n

    Quote Originally Posted by harpazo View Post
    Thanks. Give me another example to practice.
    find all integers between 1 and 100 whose LENGTH is 5
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    Senior Member harpazo's Avatar
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    Re: The "Length" of n

    Quote Originally Posted by Idea View Post
    find all integers between 1 and 100 whose LENGTH is 5
    2 2 2 2 2 = 32
    2 2 2 2 3 = 48
    2 2 2 3 3 = 72
    2 2 2 2 5 = 80

    That's all of them.
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    Re: The "Length" of n

    Quote Originally Posted by harpazo View Post
    NOTE: CAN SOMEONE MOVE THIS QUESTION TO THE ALGEBRA SECTION? I posted here by mistake.
    For any positive integer n, n > 1, the LENGTH of n is the number of positive primes (not necessarily distinct) whose product is n. For example, the length of 50 is 3 since 50 = (2)(5)(5).
    Which of the following integers has length 3?
    A. 3
    B. 15
    C. 60
    D. 64
    E. 105
    I selected 60 but the correct answer is 105.
    @harpazo, your problem is that you do not bother to understand definitions and even if you do, you absolutely have no idea how to apply them.
    By the given, the length of a positive integer is simply the sum of the exponents in the prime fraternization of the number.
    What is the length of 3872000? Well kook at it's prime factorization: SEE HERE
    It's length is the sum of the exponents. So what is the length of 3872000?

    WHAT IS THE LENGTH OF 105?
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    Re: The "Length" of n

    Quote Originally Posted by Plato View Post
    <snip> prime fraternization of the number. <snip>
    durn autocorrect!
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    Senior Member harpazo's Avatar
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    Re: The "Length" of n

    Quote Originally Posted by Plato View Post
    @harpazo, your problem is that you do not bother to understand definitions and even if you do, you absolutely have no idea how to apply them.
    By the given, the length of a positive integer is simply the sum of the exponents in the prime fraternization of the number.
    What is the length of 3872000? Well kook at it's prime factorization: SEE HERE
    It's length is the sum of the exponents. So what is the length of 3872000?

    WHAT IS THE LENGTH OF 105?
    2 2 2 2 2 = 32
    2 2 2 2 3 = 48
    2 2 2 3 3 = 72
    2 2 2 2 5 = 80

    That's all of them.
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    Re: The "Length" of n

    Quote Originally Posted by harpazo View Post
    2 2 2 2 2 = 32
    2 2 2 2 3 = 48
    2 2 2 3 3 = 72
    2 2 2 2 5 = 80
    That's all of them.
    @harpazo, do you ever read our replies?
    The prime factorization of $105=3^1\cdot 5^1\cdot 7^1$ Then $1+1+1=3$ so the length, $L(\mathit{105})=3$
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    Senior Member harpazo's Avatar
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    Re: The "Length" of n

    Good. Moving on.
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    Re: The "Length" of n

    Quote Originally Posted by harpazo View Post
    Good. Moving on.
    Moving on to what?
    Have you learned anything from any of this?
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  12. #12
    Senior Member harpazo's Avatar
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    Re: The "Length" of n

    Moving on to the next chapter.
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    Re: The "Length" of n

    Quote Originally Posted by harpazo View Post
    Moving on to the next chapter.
    Why don't you fully try to understand the "present chapter" first?!
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    Re: The "Length" of n

    Quote Originally Posted by harpazo View Post
    2 2 2 2 2 = 32
    2 2 2 2 3 = 48
    2 2 2 3 3 = 72
    2 2 2 2 5 = 80

    That's all of them.
    Yes that's correct and you've done it in an efficient way.

    I think you totally understanding the meaning of "length" of a number.
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  15. #15
    Senior Member harpazo's Avatar
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    Re: The "Length" of n

    Quote Originally Posted by Debsta View Post
    Yes that's correct and you've done it in an efficient way.

    I think you totally understanding the meaning of "length" of a number.
    Thank you for your patience and understanding. I do not reply to those who enjoy putting me down. There's a HUGE difference between constructive criticism and belittling. You have have professional with me. I thank you.
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