How Many Factors Does 36^2 Have? GMAT Problem Solving

Question: How many factors does \(36^2\) have?

1. 2
2. 8
3. 24
4. 25
5. 26

Correct Answer: D

Solution and Explanation:
Approach Solution 1:
It is important to recognize that this question is asking for all the numbers that go into the big number. Breakdown the big number into its prime factorization - put it into bases of 2, 3, 5, 7, etc...and assign exponents

Step 2:

\(36^2\) = 36 * 36

= 6 * 6 * 6 * 6
= \(6^4\)

= \((2*3)^4\)

= \( 2^4 * 3^4\)

Step 3:

Once prime factorization is applied (put into bases of 2, 3, 5, 7, etc), number of factors are found..
It turns out that the exponents can tell us how many factors are in the number. On a smaller scale, if we had the number 8, its factors are 1, 2, 4, 8.

\(8 = 2^3\)

All lower exponents of \(2^3 \)are also factors, such as\( 2^2, 2^1 and 2^0\). Thus \(2^3 \)has 3 factors + "1" which is also a factor so \( 2^3 \)has 4 factors. so, just take the exponent and add 1.
Now this was for 8, now if multiplied by a different base like 3? so 8*3 = 24, which breaks down to\( 2^3 * 3\). How many factors does 24 have?

Well 24 = 2* 2* 2* 3; the factors are basically any combination of the prime numbers. By adding the 3, we added more ways to combine the prime numbers. Now add all previous factors by 3 and create a new set of factors:\( 2^0 * 3, 2^1 *3, 2^2 * 3,\) \(2^3 * 3 = {3, 6, 12, 24}\)

in addition to {1, 2, 4, 8}; essentially multiplying the number of factors by 2. 4*2 = 8 factors now
So mathematically it looks like

\(24 = (2^3)*(3^1)\)

= (set of 4 factors) * (set of 2 factors) - multiply since we want to find the number of combinations of 2's and 3's that form new factors
= 8 factors

Step 4:

Now, going back to our big number:

\(36^2 = 2^4 * 3^4\)

The number of factors are combinations of factors of 2^4 with factors of 3^4. So, the # of factors for 2^4 multiplied by # of factors of 3^4 will give you the answer

\(2^4\) = 16; has factors \(2^0, 2^1, 2^2, 2^3, 2^4\) = 1, 2, 4, 8, 16 (or 5 factors)

\(3^4\) = 81; has factors \(3^0, 3^1, 3^2, 3^3, 3^4\) = 1, 3, 9, 27, 81 (or 5 factors)
Now combine any of the first set of factors with the second set of factors = 5*5 = 25
And take {1, 3, 9, 27, 81} and multiply by 1.
Or it can be multiplied by 2. {3, 6, 18, 54, 162} are all factors of \(36^2\).
Or it can also be multiplied by 4. By 8, By 16.
The largest number you get is 16*81 = 1296. And it's seen that it is equal to \(36^2\), so you know as a check that these two numbers multiplied still give you a factor. That factor being 1296, does go into\( 36^2\).
In total there are 5*5 = 25 combinations of different factors generated.

Approach Solution 2:
It is possible to solve the following equation and the arithmetic approach can also be useful in this case. This can be assessed in the following way:"

\(36^2 = 2^4*3^4\)

so factors = (4+1)(4+1) (power of each prime factor + 1)
=25

Hence, \(36^2\) have 25 factors.

Approach Solution 3:
The problem statment asks to find out the number of factors does \(36^2\) have.

An interesting fact about perfect squares greater than 1 is that they always have an odd number of factors.

For instance, the factors of 4 are 1, 2, and 4 (a total of 3 factors) and the factors of 100 are 1, 2, 5, 10, 20, 50, and 100 (a total of 7 factors).

Since 36^2 is a perfect square, it should have an odd number of factors.
Therefore, by analysing all the options given, we can say:
The only odd number in the answer choices is 25.

Hence, \(36^2\) have 25 factors.

“How many factors does \(36^2\) have”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This topic has been taken from the book “GMAT Advanced Quant”. The GMAT Problem Solving questions enable the candidates to evaluate information and solve numerical problems. GMAT Quant practice papers help the candidates to analyse several types of questions that will enable them to improve their mathematical knowledge.

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