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# What is the value of mn?

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What is the value of mn?  [#permalink]

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07 Dec 2019, 17:56
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GMATBusters’ Quant Quiz Question -1

What is the value of mn?

(1) $$5^m*3^n=45$$

(2) $$9^{(m-1)(n-2)}=1$$

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Re: What is the value of mn?  [#permalink]

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29 Mar 2020, 04:15
NandishSS wrote:
Quote:

GMATBusters’ Quant Quiz Question -1

What is the value of mn?

(1) $$5^m*3^n=45$$

(2) $$9^{(m-1)(n-2)}=1$$

Can you please show me how apart from m=1 & n=2 doesn't satisfy?

This is a copy of the following GMAT Club question: https://gmatclub.com/forum/new-tough-an ... l#p1029249 Below is an explanation for statement (1):
if $$x$$ and $$y$$ are integers then as $$75=3^1*5^2$$ then $$x=1$$ and $$y=2$$ BUT if they are not, then for any value of $$x$$ there will exist some non-integer $$y$$ to satisfy given expression and vise-versa (for example if $$y=1$$ then $$3^x*5^y=3^x*5=75$$ --> $$3^x=25$$ --> $$x=some \ irrational \ #\approx{2.9}$$). Not sufficient.
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Re: What is the value of mn?  [#permalink]

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07 Dec 2019, 18:18
1

1) Using prime factorization convert 45= 3^2*5

m=1 and n=2 gives mn=2

2) has multiple possibilites as 1=1^0 or 1^1
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Re: What is the value of mn?  [#permalink]

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07 Dec 2019, 19:26
Required: value of m*n?

Statement 1: 5^m * 3^n = 45

For the above equation to be valid m has to take value 1 and n has to take value 2

Hence, Sufficient.

Statement 2: 9 ^ ((m-1) * (n-2)) = 1

9^0 = 1
(m-1) * (n-2) = 0
1. m equals 1 or (not sufficient)
2. n equals 2 or (not sufficient)
3. both (sufficient)

Hence, Not sufficient

Ans A.

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Re: What is the value of mn?  [#permalink]

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07 Dec 2019, 19:49
What is the value of mn?

(Statement1): $$5^{m}∗3^{n}=45$$
if m=1, n=2 and $$5^{1}∗3^{2}=45$$, then --> m*n =1*2=2

--> but in the question stem, it does not say whether m,n --are positive integer.
if n=0, then m will be equal to $$log_5$$45 (real number)
--> m*n will be equal to 0(zero)
TWO values
Insufficient

(Statement2): $$9^{(m−1)(n−2)}=1$$
$$9^{(m−1)(n−2)}=9^{0}$$
(m−1)(n−2)=0

--> In order (m−1)(n−2) be equal to 0,
one of (m-1) and (n-2) is enough to equal to zero.

--> (case1): if m=1,n=2 and (m−1)(n−2)=0, then m*n=1*2.
--> (case1): if m=1, n=3 and (1-1)(3-2)=0*1=0, then m*n=1*3=3
TWO values
Insufficient

Taken together 1&2,
In this equation (m−1)(n−2)=0:
--> if m=1, then $$5^{1}*3^{n}= 45$$--> n= 2 (must) --> m*n=2
--> if n=2, then $$5^{m}*3^{2}= 45$$--> m=1(must) --> m*n=2

The value of m*n which is equal to 2 is UNIQUE
Sufficient

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Re: What is the value of mn?  [#permalink]

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07 Dec 2019, 20:24
45 = 5 x 3^2
Making it 5^m x 3^n as m=1and 3=2

With statement 2. (M-1)(n-2)=0
Here either m =1 and n could be anything or n =2 and m could be anything. No distinct answers.

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Re: What is the value of mn?  [#permalink]

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07 Dec 2019, 22:30
value of mn?

A) 5^m * 3^n =45

m has to be 1
n has to be 2

so mn=2.1=2

sufficient hence,A

B)9^(m-1)(n-2) =1

for 9^anything to be 1>>9^0 =1

m=1 n=2 mn=2.1=2 or m=1 n=50 mn=50.1=50

two different answers so not sufficient
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Re: What is the value of mn?  [#permalink]

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07 Dec 2019, 23:10
Q: m*n=?
(1) 3^n*5^m = 3^2*5^1
hence, n=2 and m=1 (Sufficient)
(2) 9^(m-1)(n-2) = 1
hence, (m-1)(n-2) = 0
either m=1 or n=2 (Not Sufficient)
Ans: A
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Re: What is the value of mn?  [#permalink]

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07 Dec 2019, 23:16
Since it is not given that m and n are integers

1. This option give one value for m and n if they are integer
We can also get other value for m and n , if they are not integer.

Thus 1 is not sufficient alone.

2. With 2 alone also we get that if value of m is 1, value of n can be anything.
If value of n is 2, value of m can be anything.

So 2 alone is also not sufficient

Taken 1 and 2 together is sufficient. As through option 2 we get value of m is 1 or the value of n is 2; and correspondingly we get value of n or m from option 1

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Re: What is the value of mn?  [#permalink]

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Updated on: 08 Dec 2019, 19:15
value of m*n
#1
45= 3^2*5^1
so m=1 and n=2 ; mn= 2
sufficient
Or 15^1*3^1 mn =1 insufficient
#2
9^(m−1)(n−2)=1
possible 9^0
so either m=1 or n=2
but value of m and n can be different if either of them is m=1 or n=2 or could be same as well
m*n could vary
insufficient
From 1&2
M =1 n=2
IMO c

What is the value of mn?

(1) 5m∗3n=45

(2) 9^(m−1)(n−2)=1

Originally posted by Archit3110 on 07 Dec 2019, 23:50.
Last edited by Archit3110 on 08 Dec 2019, 19:15, edited 1 time in total.
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Re: What is the value of mn?  [#permalink]

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07 Dec 2019, 23:52
Let us look at each statements one by one

(1) 5^m∗3^n=45

Here only possible values of m and n which satisfies the equation are m = 1 and n =2 Hence statement 1 alone is sufficient

(2) 9^(m−1)(n−2)=1

As per the statement, the possible values of m and n are 1 m =1 and n=2 , Hence statement 2 alone is also sufficient

So Final Answer is Option D Each Statement Alone is sufficient
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Re: What is the value of mn?  [#permalink]

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08 Dec 2019, 00:04
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What is the value of mn?

Many would surely go wrong by answering A because you would be taking m and n as integers

(1) $$5^m*3^n=45=5*3^2$$
we can take mn=1*2=2, but you take m as 3, there will be some value of n that will satisfy the given equation.
We do not have to find more values but knowing this is enough.

(2) $$9^{(m-1)(n-2)}=1$$
either m=1, n=2 or both

Combined..
we know at least one of the two m=1 or n=2 is true.
Substituting any of the value in statement I would also satisfy the other value, so m=1 and n=2, and mn =2

C
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Re: What is the value of mn?  [#permalink]

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08 Dec 2019, 01:06
What is the value of mn?

(1) 5^m∗3^n=45,
5^m∗3^n = 3^2∗5^1
so, m = 1 and n=2,
mn= 2
Hence, sufficient.

(2) 9^(m−1)(n−2)=1
3^2(m-1)(n-2) = 1
3^(2m-2)(2n-4) = 1
3^(4mn-4n-8m+8) = 1
One equation with two variables can have multiple number possibilities for mn. Hence, insufficient.

Imo. A.
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Re: What is the value of mn?  [#permalink]

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08 Dec 2019, 02:32

Explanation:

FROM A:
Given $$5^m*3^n=45$$

For equations with exponents on both sides, we can eliminate base if we can keep the bases same on both sides. This way, we can equate the exponents value of the common bases.

So, in our approach, we try to keep the base same here.

We can express 45 as $$3^2*5^1$$

$$5^m*3^n=3^2*5*1$$

Same integers value would have same exponents.
This means, m = 1 (common base 5), and n=2 (common base 3).

Hence, from A, we can get the definite answer of m*n = 1*2 = 2. Eliminate BCE.

FROM B:
We can express 1 as 9^0, which is equal to 1.

$$9^(m-1)(n-2)=9^0$$

This gives us, (m-1)(n-2)=0 (common base 9)
Now, if product of two integers are equal to zero, either one of them, or both of them, are zero.
So either m = 1, then we can't get value of n, or n = 2, then we can't get value of m, or both m-1 = 0, and n-2 = 0, and we get value of m and n.

So we are not getting definite answer of mn. Hence, we eliminate D.

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Re: What is the value of mn?  [#permalink]

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08 Dec 2019, 03:47
we need to find nm=?

(1) 5m∗3n=45= 3^2*5^1
comparing indices, we get m=1, n=2 hence mn=2

(2) 9^(m−1)(n−2)=1
in this case-
(m-1)(n-1) = 0
so if (m-1)=0 then (n-1) may or may not equal to zero, giving different n values

similarly if (n-1)=0 then (m-1) may or may not equal to zero, hence may give different values of n.
hence this is insufficient.
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Re: What is the value of mn?  [#permalink]

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08 Dec 2019, 05:31
Since that m and n are integers is not given, we can't conlude anything from the first equation. Given the second equation, we know that either m = 1 or n = 2. Combined with (1), each case returns the same pair of (m, n) -> mn = 2, but it takes both equations to come to this
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Re: What is the value of mn?  [#permalink]

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Updated on: 08 Dec 2019, 21:03
(1) (5^m)∗(3^n) = 45 …(i)
45 = 3^2*5^1 …(ii)
From the above two, we get m=1, n=2.

Also, for m=0, we will get 3^n = 45 for some real value of n.
Therefore, we do not have a unique solution.

(2) 9^{(m-1)(n-2)} = 1 …(i)
9^{(m-1)(n-2)} = 9^0 …(ii)
Comaparing (i) and (ii), we get (m-1)(n-2) = 0
Therefore, either
m-1 = 0 or
n-2 = 0
For m=1, n can take any real value and still satisfy (i). Therefore, we do not have a unique solution.
Similarly, for n=2, m can take any real value and still satisfy (ii). Therefore, we do not have a unique solution.

Combining statements 1 and 2 we get m = 1 and n = 2 which is a unique solution.
Therefore, Choice C is the answer.

Originally posted by jhavyom on 08 Dec 2019, 05:39.
Last edited by jhavyom on 08 Dec 2019, 21:03, edited 1 time in total.
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Re: What is the value of mn?  [#permalink]

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08 Dec 2019, 07:54
What is the value of mn?

(1) 5m∗3n=455m∗3n=45

(2) 9(m−1)(n−2)=1

Let's assess option 1. the Prime Factorization of 45 is 3^2 and 5.
Hence, we can find the unique solution for m and n. Now need to check for option 2

Option 2, result of any numbers raised to the power of 0 is zero. From here, (m-1)(n-2) = 0
However, there are many solutions possible. (E.g. m=1 and n=3 or m=2 and n=2). Hence, this option only is not sufficient to answer

In conclusion, option 1 only is suffice to answer, but option 2 only is not sufficient (A)
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Re: What is the value of mn?  [#permalink]

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08 Dec 2019, 11:35
ANS-C, Statement A will be sufficient if m and n are integer.
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Re: What is the value of mn?  [#permalink]

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09 Dec 2019, 03:57
(1) (5^m)∗(3^n) = 45
Since m and n are not integers, they can take any real value.
Not sufficient.

(2) 9^{(m-1)(n-2)} = 1
9^{(m-1)(n-2)} = 9^0
From the above two equations, (m-1)(n-2) = 0
Therefore, either
m-1 = 0 or n-2 = 0
For m=1, n can take any real value.
For n=2, m can take any real value.

Combining (1) and (2)
m = 1, n = 2
Unique solution.
Re: What is the value of mn?   [#permalink] 09 Dec 2019, 03:57

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