How Many Factors of \(72^2\) Are Multiples of 6? GMAT Problem Solving

Question: How many factors of \(72^2\) are multiples of 6?

1. 35
2. 25
3. 30
4. 24
5. 12

Correct Answer: D

Solution and Explanation
Approach Solution 1:

Prime factorization can be defined as a way of expressing a number as a product derived from its prime factors. A prime number is a number that has only two factors, 1 and the number itself.
Accordingly, to find the number of factors that \(72^2\) has, it is important to evaluate through prime factorisation the values as follows:

Prime factorisation of \((72)^2\) = \((36 • 2)^2\) = \((62 • 2)^2\)= \((2^23^2 • 2)^2\) =\(2^63^4\)

Significantly, the number of factors of an integer is said to be the product of the "power+1" of its prime factors:

This typically implies for \(2^53^3\) = (5 + 1)(3 + 1)

The result is equal to 24.
Accordingly, in order to get the answer for the given case in terms of the number of factors which are multiples of 6 of \(72^2\), there are 24 factors.
Thus, D is the correct answer.

Approach Solution 2:

Considering that factorisation of \(72^2\) is important, it may be identified that the evaluation of the factorization of \(72^2\). To find the number of factors can be as follows:

\(72^2 = (2^33^2)^2 = 2^63^4\)

From here, 6 can be evaluated as a number with factors 2 and 3. Accordingly, using the multiples of 2 and 3, the number of factors can be determined as (6 +1)(4 +1)
This implies that 7 multiplied with 5 would generate the result 12.
However, in each case the factor 1 has been added twice for 2 and 3, which can further be implicated as
12 - 1 = 11
Hence, the number of factors which stand as multiples of 6 include 35 - 11 which is equal to 24.
Hence, option D is the correct answer.

Approach Solution 3:
Prime factorization of (72)2=(36∙2)2=(62∙2)2=(2232∙2)2=2634(72)2=(36•2)2=(62•2)2=(2232•2)2=2634
num factors of an integer is the product of the "power+1" of its prime factors:
722=2634…6=2∙3…m(6)=722=(2∙3)(2533)…num.factors=(2533)=(5+1)(3+1)=24722=2634…6=2•3…m(6)=722=(2•3)(2533)…num.factors=(2533)=(5+1)(3+1)=24

Let us factorization 72^2..
72^2= (2^33^2)^2= 2^63^4
Number of factors = (6+1)(4+1)=7*5=35
Now 6 is 2*3, so just multiple of just 2 and 3 are (6+1) and (4+1), so total 7+5=12..
But in each case, we have added factor 1 twice, once with 2 and once with 3..
So 12-1=11...
Number of factors that are multiples of 6 are 35-11=24..

“How many factors of \(72^2\) are multiples of 6?”- is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.

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