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If 20/5 = 1/2^m + 1/2^n what is nm ?

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If 20/5 = 1/2^m + 1/2^n what is nm ?  [#permalink]

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New post Updated on: 20 Feb 2019, 02:21
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If \(\frac{20}{2^{5}} = \frac{1}{2^{m}} + \frac{1}{2^{n}}\) what is nm ?

a) 0
b) 3
c) 8
d) 16
e) 24

Originally posted by Pdirienzo on 19 Feb 2019, 11:05.
Last edited by Pdirienzo on 20 Feb 2019, 02:21, edited 1 time in total.
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Re: If 20/5 = 1/2^m + 1/2^n what is nm ?  [#permalink]

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New post 19 Feb 2019, 14:03
Guys....anyone knows..how to solve this..appreciate responses
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Re: If 20/5 = 1/2^m + 1/2^n what is nm ?  [#permalink]

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New post 19 Feb 2019, 15:40
1
Pdirienzo wrote:
If \(\frac{20}{5} = \frac{1}{2^{m}} + \frac{1}{2^{n}}\) what is nm ?
a) 0
b) 3
c) 8
d) 16
e) 24


I didn't get the algebraic key to solve it, but I had some thoughts.

when we think about the idea that the sum of two fractions is 4,
this means that at least m or n must be negative to convert the fraction to an integer.
Also we can conclude that if one of the variables (m or n) is negative, the other can't be positive because it will give a mixed number upon summation.
so the options of m and n are either (-ve,-ve) or (-ve,0).

by looking at the value of 4, it is an even number.
so (-ve,0) is excluded because it would result in an odd number (because 2^0 = 1, which is odd).

by thinking about the (-ve,-ve) option, I though of the pair (-1,-1), which would make mn = 1, but it is not in the answer choices :lol:
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If 20/5 = 1/2^m + 1/2^n what is n?  [#permalink]

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New post Updated on: 20 Feb 2019, 02:47
20/(2^5)=1/2^m + 1/2^n
20/(2^5)=2^n+2^m/(2^m*2^n)
20(2^m*2^n)=(2^n+2^m)2^5
2^2*5(2^m*2^n)=(2^n+2^m)2^5
5(2^m*2^n)=(2^n+2^m)2^3
Now the only way the LHS will equate the RHS is when m=3 ,n=1 or when n=3 ,m=1
5(2^3*2^1)=(2^1+2^3)2^3
Now ,5*2=(2^1+2^3)
So m*n =3
Ans is B


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Originally posted by Staphyk on 20 Feb 2019, 00:36.
Last edited by Staphyk on 20 Feb 2019, 02:47, edited 1 time in total.
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Re: If 20/5 = 1/2^m + 1/2^n what is nm ?  [#permalink]

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New post 20 Feb 2019, 02:22
Sorry, there was a mistake in the question, already edited the original
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Re: If 20/5 = 1/2^m + 1/2^n what is nm ?  [#permalink]

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New post 20 Feb 2019, 02:47
Staphyk wrote:
(2^2*2^m*2^n)=2^n+2^m
2^2+m+n= 2(1^n+1^m)

Hi Staphyk
In General, taking 2 as a common factor is not a valid simplification step.
Think about it. \(2^3 + 2^2 \neq{2(1^3+1^2)}\)
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Re: If 20/5 = 1/2^m + 1/2^n what is nm ?  [#permalink]

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New post 20 Feb 2019, 03:14
2
Here are my two cents.

20/32 = 5/8

= (1+4)8 = 1/8 +4/8
= 1/8 + 1/2

= 1/2^3 + 1/2^1

SO by comparing both side

we get m=3 and n=1

so mn = 3

I could not write full soultion in this due to typing overhead.

Please award kudos if helpful

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Re: If 20/5 = 1/2^m + 1/2^n what is nm ?  [#permalink]

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New post 20 Feb 2019, 03:34
20/2^5=2^m+2^n/2^mn
4*5/2^5=2^m+2^n/2^mn
5/2^3=2^m+2^n/2^mn
so all individual values of m and n, mn should always be 3 to satisfy the equation.

3 Kudos short.

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Re: If 20/5 = 1/2^m + 1/2^n what is nm ?  [#permalink]

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New post 21 Feb 2019, 18:19
1
Pdirienzo wrote:
If \(\frac{20}{2^{5}} = \frac{1}{2^{m}} + \frac{1}{2^{n}}\) what is nm ?

a) 0
b) 3
c) 8
d) 16
e) 24


(Note: Here we are assuming that both m and n are integers.)

Simplifying, we have:

20/(2^5) = 1/(2^m) + 1/(2^n)

5/2^3 = 1/(2^m) + 1/(2^n)

5 = (2^3)/(2^m) + (2^3)/(2^n)

5 = 2^(3 - m) + 2^(3 - n)

The only way to express 5 as the sum of two integer powers of 2 is 5 = 4 + 1 = 2^2 + 2^0. So, either of (3 - m) or (3 - n) is equal to 2 and the other one is equal to 0. If 3 - m = 2 and 3 - n = 0, then m = 1 and n = 3. In this case, we get mn = 3. If 3 - m = 0 and 3 - n = 2, we get m = 3 and n = 1 and again, mn = 3.

Answer: B
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If 20/5 = 1/2^m + 1/2^n what is nm ?  [#permalink]

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New post 23 Feb 2019, 07:30
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\(\frac{20}{2^5} = \frac{1}{2^m} + \frac{1}{2^n}\)

\(\frac{5}{2^3} = \frac{1}{2^m} + \frac{1}{2^n}\)

\(\frac{(4 +1)}{2^3} = \frac{1}{2^m} + \frac{1}{2^n}\)

\(\frac{(2^2+1)}{2^3} = \frac{1}{2^m} + \frac{1}{2^n}\)

\(\frac{1}{2^1} + \frac{1}{2^3} = \frac{1}{2^m} + \frac{1}{2^n}\)

this implies m = 1 when n = 3 or m = 3 when n = 1

thus mn = 3
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If 20/5 = 1/2^m + 1/2^n what is nm ?   [#permalink] 23 Feb 2019, 07:30
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