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Math Expert V
Joined: 02 Sep 2009
Posts: 65062
The greatest common factor of positive integers m and n is 12. What is  [#permalink]

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23 00:00

Difficulty:   35% (medium)

Question Stats: 66% (01:23) correct 34% (01:27) wrong based on 474 sessions

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The greatest common factor of positive integers m and n is 12. What is the greatest common factor of (2m^2, 2n^2)?

A. 2
B. 12
C. 24
D. 144
E. 288

Kudos for a correct solution.

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Re: The greatest common factor of positive integers m and n is 12. What is  [#permalink]

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2
Bunuel wrote:
The greatest common factor of positive integers m and n is 12. What is the greatest common factor of (2m^2, 2n^2)?

A. 2
B. 12
C. 24
D. 144
E. 288

Kudos for a correct solution.

METHOD-1

Given : GCD of (m and n) = 12 = 2^2*3

i.e. m and n are both multiples of 2^2*3

i.e. m^2 and n^2 will both be multiples of (2^2*3)^2 = 2^4*3^2

i.e. 2m^2 and 2n^2 will both be multiples of 2(2^2*3)^2 = 2^5*3^2 = 288

METHOD-2

Let, m = 12 and n = 24
i.e. GCD of m and n = 12

2m^2 = 288
2n^2 = 1152
i.e. GCD of 2m^2 and 2n^2 = 288

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The greatest common factor of positive integers m and n is 12. What is  [#permalink]

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5
Bunuel wrote:
The greatest common factor of positive integers m and n is 12. What is the greatest common factor of (2m^2, 2n^2)?

A. 2
B. 12
C. 24
D. 144
E. 288

Kudos for a correct solution.

GCD (m,n)=12

Thus $$m = 2^2*3*p$$
and $$n = 2^2*3*q$$, with p,q co-primes

Now $$2m^2 = 2^5*3^2*p^2$$
and $$2n^2 = 2^5*3^2*q^2$$

Thus GCD ($$2m^2, 2n^2$$) = $$2^5*3^2$$ = 288. Thus E is the correct answer.
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Math Expert V
Joined: 02 Sep 2009
Posts: 65062
Re: The greatest common factor of positive integers m and n is 12. What is  [#permalink]

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2
Bunuel wrote:
The greatest common factor of positive integers m and n is 12. What is the greatest common factor of (2m^2, 2n^2)?

A. 2
B. 12
C. 24
D. 144
E. 288

Kudos for a correct solution.

800score Official Solution:

Suppose we factorize m and n into prime factors. The greatest common factor of positive integers m and n is 12 so the only prime factors m and n have in common are 2, 2 and 3. (12 = 2 × 2 × 3).

If we factorize m² into prime factors we will get each of prime factors of m twice. The same happens to n². So the only prime factors m² and n² would have in common are 2, 2, 2, 2 and 3, 3.

If we factorize 2m² into prime factors we will get the same prime factors as for m² and one more prime factor “2”. The same happens to n². So the only prime factors 2m² and 2n² would have in common are 2, 2, 2, 2, 2 and 3, 3. By factoring those we get the greatest common factor of 2m² and 2n².

2 × 2 × 2 × 2 × 2 × 3 × 3 = 288.

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Re: The greatest common factor of positive integers m and n is 12. What is  [#permalink]

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Hello,

I have a question. Can we also use the relationship between LCM and GCF to find the solution?

So, we would say:
m*n =12*x, where x is the LCM. Then,
m^2*n^2 = (12x)^2
m^2*n^2 = (12x)^2 = 144x^2. Finally,

2(m)^2 * 2 (n)^2 = 2 (144x^2)
2(m)^2 * 2 (n)^2 = 228*2x^2.

So, we end up with 228, which is E.

Is this correct?
Senior SC Moderator V
Joined: 22 May 2016
Posts: 3938
The greatest common factor of positive integers m and n is 12. What is  [#permalink]

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1
1
Bunuel wrote:
The greatest common factor of positive integers m and n is 12. What is the greatest common factor of (2m^2, 2n^2)?

A. 2
B. 12
C. 24
D. 144
E. 288

Kudos for a correct solution.

Kinda dorky, but I'm all for simple if it works. I just wrote out, in stages, what factors m and n had to have.

LCM of m and n is 12
12 = 2 * 2 * 3

m: 2, 2, 3
n: 2, 2, 3

Variables squared? Just take the prime factors and list them again:

m = (2 * 2 * 3)
m$$^2$$ = (m * m) =
(2 * 2 * 3) * (2 * 2 * 3)

After squaring both m and n we have:

m$$^2$$: 2, 2, 3, 2, 2, 3
n$$^2$$: 2, 2, 3, 2, 2, 3

Then each term * 2? (The last step. We must find 2m$$^2$$ and 2n$$^2$$)

2m$$^2$$: 2, 2, 3, 2, 2, 3, 2
2n$$^2$$: 2, 2, 3, 2, 2, 3, 2

Both have 2$$^5$$, 3$$^2$$. (32 * 9) = 288

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The greatest common factor of positive integers m and n is 12. What is  [#permalink]

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Bunuel wrote:
The greatest common factor of positive integers m and n is 12. What is the greatest common factor of (2m^2, 2n^2)?

A. 2
B. 12
C. 24
D. 144
E. 288

Kudos for a correct solution.

if GCF of m and n is 12 i.e. $$\frac{m+n}{12}$$ ---->$$\frac{2m^2}{GCF} + \frac{2n^2}{GCF}$$

$$\frac{(2m^2 + 2n^2)}{GCF}$$

factorize $$\frac{2 (m^2 + n^2)}{GCF}$$

hence $$GCF = 2(m^2)= 2(12^2) = 2*144 = 288$$

is my reasoning correct ? have a fantastic day everyone Target Test Prep Representative V
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Re: The greatest common factor of positive integers m and n is 12. What is  [#permalink]

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1
Bunuel wrote:
The greatest common factor of positive integers m and n is 12. What is the greatest common factor of (2m^2, 2n^2)?

A. 2
B. 12
C. 24
D. 144
E. 288

Notice that 2m^2 = 2 x m x m and 2n^2 = 2 x n x n. We see that the greatest factor (gcf) of (2, 2) = 2, the gcf of first pair of (m, n) = 12 and the gcf of the second pair of (m, n) = 12. Therefore, the gcf of (2m^2, 2n^2) = 2 x 12 x 12 = 288.

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Re: The greatest common factor of positive integers m and n is 12. What is  [#permalink]

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GMATinsight wrote:
Bunuel wrote:
The greatest common factor of positive integers m and n is 12. What is the greatest common factor of (2m^2, 2n^2)?

A. 2
B. 12
C. 24
D. 144
E. 288

Kudos for a correct solution.

METHOD-1

Given : GCD of (m and n) = 12 = 2^2*3

i.e. m and n are both multiples of 2^2*3

i.e. m^2 and n^2 will both be multiples of (2^2*3)^2 = 2^4*3^2

i.e. 2m^2 and 2n^2 will both be multiples of 2(2^2*3)^2 = 2^5*3^2 = 288

METHOD-2

Let, m = 12 and n = 24
i.e. GCD of m and n = 12

2m^2 = 288
2n^2 = 1152
i.e. GCD of 2m^2 and 2n^2 = 288

In this question, you know the GCD of m and n is 12, so to find GCD for 2m^2 and 2n^2, why not just square 12 and double it? So 12*12 = 144, 144*2= 288? Isn't this an easier way to obtain the answer?
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Re: The greatest common factor of positive integers m and n is 12. What is  [#permalink]

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1
Bunuel wrote:
The greatest common factor of positive integers m and n is 12. What is the greatest common factor of (2m^2, 2n^2)?

A. 2
B. 12
C. 24
D. 144
E. 288

Kudos for a correct solution.

One of the points that may concern many would be the presence of other prime factors that $$m$$ and $$n$$ would contain that we have no clue about. Let's simulate that scenario by adding a couple of unknown prime factors and understand whether those unknown prime factors actually matter at all.

We are given that the HCF/GCD of $$m$$ and $$n$$ is 1$$2$$ which means that the only common prime factors between $$m$$ and $$n$$ are two 2's and one 3 ($$12 = 2 * 2* 3$$)

Say,
$$m = 2 * 2 * 3 * 5 * 7 * 17$$
$$n = 2 * 2 * 3 * 13 * 11$$

The GCD of $$m$$ and $$n$$ from the above examples is 12. Now lets square both $$m$$ and $$n$$ as our ultimate goal is to find the GCD of $$2m^2$$ and $$2n^2$$.
$$m^2 = (2 * 2 * 3 * 5 * 7)^2 = 2^2 * 2^2 * 3^2 * 5^2 * 7^2 * 17^2$$
$$n^2 = (2 * 2 * 3 * 13 * 11)^2 = 2^2 * 2^2 * 3^2 * 13^2 * 11^2$$

The GCD of $$m^2$$ and $$n^2$$ from the above examples is $$2^2 * 2^2 * 3^2$$ which is $$12^2$$. This shows that in GCD (where we take the lowest power of the common primes) the prime factors that are NOT common do not matter.

When we multiply $$2$$ to $$m^2$$ and $$n^2$$ we get another prime factor that is common to both $$m^2$$ and $$n^2$$

Hence GCD of $$2m^2$$ and $$2n^2$$ is $$2 * 2^2 * 2^2 * 3^2 = 2^5 * 3^2 = 32 * 9 =288$$

Ans E
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Re: The greatest common factor of positive integers m and n is 12. What is  [#permalink]

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Bunuel wrote:
The greatest common factor of positive integers m and n is 12. What is the greatest common factor of (2m^2, 2n^2)?

A. 2
B. 12
C. 24
D. 144
E. 288

Kudos for a correct solution.

Given: The greatest common factor of positive integers m and n is 12.

Asked: What is the greatest common factor of (2m^2, 2n^2)?

gcd (m,n) = 12
gcd (m^2,n^2) = 144
gcd (2m^2,2n^2) = 288

IMO E
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Re: The greatest common factor of positive integers m and n is 12. What is  [#permalink]

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12 is the greatest common factor of m and n.
It means that m = (12*x) and n= (12*x).

2m^2 = 2(12*x)^2
= 2(144x^2)
= 288x^2

2n^2 = 2(12*x)^2
= 2(144x^2)
= 288x^2

For m and n = 288 * x^2
It means that the greatest common factor is 288. Re: The greatest common factor of positive integers m and n is 12. What is   [#permalink] 30 Mar 2020, 01:50

# The greatest common factor of positive integers m and n is 12. What is   