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For any positive integer n, the length of n is defined as th [#permalink]

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11 Feb 2010, 16:40

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For any positive integer n, the length of n is defined as the number of prime factors whose product is n. For example, the length of 75 is 3, since 75 = 3 * 5 * 5. How many two-digit positive integers have length 6?

For any positive integer n, the length of n is defined as the number of prime factors whose product is n. For example, the length of 75 is 3, since 75 = 3 * 5 * 5. How many two digit positive integers have length 6?

A. None B. One C. Two D. Three E. Four

Basically the length of the integer is the sum of the powers of its prime factors.

Length of six means that the sum of the powers of primes of the integer (two digit) must be \(6\). First we can conclude that \(5\) can not be a factor of this integer as the smallest integer with the length of six that has \(5\) as prime factor is \(2^5*5=160\) (length=5+1=6), not a two digit integer.

The above means that the primes of the two digit integers we are looking for can be only \(2\) and/or \(3\). \(n=2^p*3^q\), \(p+q=6\) max value of \(p\) and \(q\) is \(6\).

Let's start with the highest value of \(p\): \(n=2^6*3^0=64\) (length=6+0=6); \(n=2^5*3^1=96\) (length=5+1=6);

\(n=2^4*3^2=144\) (length=4+2=6) not good as 144 is a three digit integer.

With this approach we see that actually \(5<=p<=6\).

Re: 700 Algrbra! Need help again. Thanks so much! [#permalink]

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11 Feb 2010, 17:23

That's brilliant!!! I especially love the part where I could take 5 away. This really save tons of time! Thanks!! BTW, Thanks so much for the prompt response!

Re: 700 Algrbra! Need help again. Thanks so much! [#permalink]

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12 Feb 2010, 02:06

Bunuel wrote:

Basically the length of the integer is the sum of the powers of its prime factors.

Length of six means that the sum of the powers of primes of the integer (two digit) must be \(6\). First we can conclude that \(5\) can not be a factor of this integer as the smallest integer with the length of six that has \(5\) as prime factor is \(2^5*5=160\) (length=5+1=6), not a two digit integer.

The above means that the primes of the two digit integers we are looking for can be only \(2\) and/or \(3\). \(n=2^p*3^q\), \(p+q=6\) max value of \(p\) and \(q\) is \(6\).

Let's start with the highest value of \(p\): \(n=2^6*3^0=64\) (length=6+0=6); \(n=2^5*3^1=96\) (length=5+1=6);

\(n=2^4*3^2=144\) (length=4+2=6) not good as 144 is a three digit integer.

With this approach we see that actually \(5<=p<=6\).

For any positive n, the length of n is defined as the number of prime factors whose product is n. for example, the length of 75 is 3, since 75= 3x5x5. How many two-digit positive integers have length 6?

For any positive integer n, the length of n is defined as th [#permalink]

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29 Aug 2017, 18:21

YTT wrote:

For any positive integer n, the length of n is defined as the number of prime factors whose product is n. For example, the length of 75 is 3, since 75 = 3 * 5 * 5. How many two-digit positive integers have length 6?

A. None B. One C. Two D. Three E. Four

Lets start with smallest prime number \(2\).

\(2^6 = 64\) ---------- (Length \(= 6\))

\(2^7\) is three digit number hence cannot be \(n\).

For any positive integer n, the length of n is defined as the number of prime factors whose product is n. For example, the length of 75 is 3, since 75 = 3 * 5 * 5. How many two-digit positive integers have length 6?

A. None B. One C. Two D. Three E. Four

We need to determine how many 2-digit integers have a length of 6, or in other words, how many 2-digit integers are made up of 6 prime factors, not necessarily distinct. Let’s start with the smallest possible numbers:

2^6 = 64 (has a length of 6)

2^5 x 3^1 = 96 (has a length of 6)

Since 2^4 x 3^2 = 144 and 2^5 x 5^1 = 160 are greater than 99, there are no more 2-digit numbers that have a length of 6.

Answer: C
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