For Any Positive Integer n, the Length of n is Defined as GMAT Problem Solving

Question: 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?

None
One
Two
Three
Four

“For any positive integer n, the length of n is defined as”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “The Official Guide for GMAT Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:
Approach Solution 1:
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=2p∗3q, 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.

Correct Answer: C

Approach Solution 2:
Let's first find the smallest value with length 6.
This is the case when each prime factor is 2.
We get 2x2x2x2x2x2 = 64. This is a 2-digit positive integer. PERFECT

To find the next largest number with length 6, we'll replace one 2 with a 3
We get 3x2x2x2x2x2 = 96. This is a 2-digit positive integer. PERFECT

To find the third largest number with length 6, we'll replace another 2 with a 3
We get 3x3x2x2x2x2 = 144. This is a 3-digit positive integer. NO GOOD
So there are only two, two-digit positive integers with length 6.

Correct Answer: C

Approach Solution 3:
According to the question,
n=a^p∗b^q∗c^r
Length = p+q+r

We need to find two-digit positive integers whose length=6; means, p+q=6

Case 1)
2^6; length=6; n=64

Case 2)
2^5∗3^1; length=6; n=96

Case 3)
2^4∗3^2; length=6; n=144 Three-Digit Incorrect

Case 5)
2^3∗3^3; length=6; n=216 Three- Digit Incorrect
As we increase, the value of n will increase beyond 2-digit positive integer

Correct Answer: C

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