2^m+2019 = |n-2020|+n-2020, where m and n are non-negative integers an

Question

2^m+2019 = |n-2020|+n-2020

, where m and n are non-negative integers and satisfy the above given equation. Find the value of (m+n)?

A. 0
B. 2019
C. 2020
D. 3030
E. 4049

ArunSharma12 · Accepted Answer

2^m+2019 = |n-2020|+n-2020 
n can not be less than 2020, as it leads to
 2^m = -2019  and 2^m can not be a negative value.

n > 2020
 2^m+2019 = n-2020+n-2020 
 2^m+6059 = 2n 
 2n - 2^m= 6059 
2^m has to be odd in order to the difference to an odd value
m = 0.
 2n = 6060 ; n = 3030

Ans: D

gurmukh · Answer

If n=3030 
Then RHS
|3030-2020|+ 3030-2020
1010 +1010
=2020
So,
2^m +2019= 2020
2^m= 1
m=0
m+n =3030
Option D is the answer

Posted from my mobile device

chetan2u · Answer

2^m+2019 = |n-2020|+n-2020

Two possibilities

1)  n\leq{2020} 
 2^m+2019=2020-n+n-2020=0 ...NOT possible

2)  n>2020 
 2^m+2019 = |n-2020|+n-2020=n-2020+n-2020=2n-4040=2(n-2020) 
Now  2^m+2019  will be even only when m is 0... .2^0+2019=2020=2(n-2020)......n-2020=1010...n=3030

m+n=0+3030=3030

D

CrackverbalGMAT · Answer

A question like this is tricky because it doesn’t test you on rules, it tests you on exceptions to that rule.
\(2^m\) is perceived by a lot of test takers to be always even. The exception is \(2^0\) which is 1 and is odd. A clue to this effect is given by telling you that m and n are non-negative integers –> note that 0 is non-negative.

The RHS can be simplified in two ways by using the definition of modulus.

|n-2020| = n-2020 if n≥ 2020
|n-2020| = 2020 – n if n<2020.

When we substitute the second case, we have \(2^m\) + 2019 = 2020 – n + n -2020. Simplifying, we have \(2^m\) + 2019 = 0 or \(2^m\) = -2019.
No integral power of 2 can yield -2019. As such, we can rule out the possibility that n<2020.

If n≥ 2020, \(2^m\) + 2019 = n-2020 + n – 2020. Simplifying, we have,

\(2^m\) + 2019 = 2n – 4040 OR

\(2^m\) = 2n – 6059.

2n is even and 6059 is odd. Even – Odd = Odd.

This means \(2^m\) is odd. This is possible only in one case –> when m = 0.

Substituting this value of m = 0 in the equation and simplifying, we have 2n = 6059 + 1 which yields us n = 3030.
The value of m+n = 3030 + 0 = 3030.

The correct answer option is D.

Hope that helps!

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

2^m+2019 = |n-2020|+n-2020, where m and n are non-negative integers an

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

2^m+2019 = |n-2020|+n-2020, where m and n are non-negative integers an

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards