For positive integers n and m (nm), is the average (arithmetic mean)

Question

For positive integers n and m  n>m , is the average (arithmetic mean) of  4*10^n ,  4*10^{n-1} , ......, and  4*10^{n-m}  an integer?

1)  m<6 
2)  n=10

chetan2u · Accepted Answer

Hi

Few important points to solve this Q..
 1) n>m, so lowest term  4*10^{n-m}  will be ATLEAST 4*10 or 40
2) Therefore each term will be multiple of 40 or 2,4,5,8,10,20,40
3) Average of the terms is " INTEGER or Not" depends on number of elements.
4) number of elements depends on 'm' and is m+1
5) since each term multiple of 40, so if m is multiple of the factors of 40, answer is YES, otherwise it has to be checked

Let's see the statements:-
1) m<6
Therefore m can be 1 to 5
So number of terms can be 1 to 5+1..
We already know if number of elements are factors of 40, and is YES..
So 1,2,4,5 will have answer as YES..
Let's check for 3 and 6..

Each number in series consists of digits 4 and 0.... 40,400,4000 etc
So if there are 3 or 6 numbers, the SUM of digits of total of these numbers will be 3*4 or 6*4, which will be divisible by 3, thus we will have an integer here too..
Example:- 40+400+4000=4440.. sum of digits is 12, so number is div by 3 
Or 400000+40000+4000=444000, again SUM of digits is 12

6 numbers will have total div by 3 and the numbers being EVEN, total will be div by 6, thus an integer again

Ans is YES always 
Sufficient

2) n=10..
We are concerned about m only
Insufficient

A

arunavamunshi1988 · Answer

Hi Chetan,

I did not understand your explanation.

himanshu1105 · Answer

Ques: Arithmetic mean an integer? 
we can simply take common from the expression 4∗10^n , 4∗10^n−1, ......, and 4∗10^n−m
=> 4*10^n (1+1/10 +1/10^2+......+1/10^m)---------(1)

so here we can see that the value of 'n' doesn't matter...it can be 40,400,40000.....
The only information you need is the value of 'm' , hence A sufficient !

Further: you can take the LCM of equation 1
4*10^n-m (10^m+10^m-1.....+1) => 4*10^n-m (11111.....m times)

Madhavi1990 · Answer

How did we assume m to be between 1 to 5? if M< 6 then can't m be negative, like say m = -6?

ShankSouljaBoi · Answer

Stem mentions n and m are positive integers

Kinshook · Answer

Asked: For positive integers n and m  n>m , is the average (arithmetic mean) of  4*10^n ,  4*10^{n-1} , ......, and  4*10^{n-m}  an integer?
Number of terms = n - (n-m) +1 = m+1
the average (arithmetic mean) of  4*10^n ,  4*10^{n-1} , ......, and  4*10^{n-m}  =  \frac{4*10^n + 4*10^{n-1} + ......+4*10^{n-m}}{(m+1)}

1)  m<6 
Let us take m=0
 4*10^n/1 = 4*10^n  Integer
Let us take m=1
 \frac{4*10^n + 4*10^{n-1}}{2} = \frac{4*10^{n-1} ( 1+10)}{2} =\frac{4*10^{n-1} *11}{2}  Integer
Let us take m=2
 \frac{4*10^n + 4*10^{n-1} + 4*10^{n-2}}{3} = \frac{4*10^{n-2} ( 1+10+100)}{3} =\frac{4*10^{n-2} *111}{3}  Integer
Let us take m=3
 \frac{4*10^n + 4*10^{n-1} + 4*10^{n-2} + 4*10^{n-3}}{4} = \frac{4*10^{n-3} ( 1+10+100+1000)}{4} = \frac{4*10^{n-3} *1111}{4}  Integer
Let us take m=4
 \frac{4*10^n + 4*10^{n-1} + 4*10^{n-2} + 4*10^{n-3} + 4*10^{n-4}}{5} = \frac{4*10^{n-4} ( 1+10+100+1000+10000)}{5} = \frac{4*10^{n-4} *11111}{5}  Integer
Let us take m=5
 \frac{4*10^n + 4*10^{n-1} + 4*10^{n-2} + 4*10^{n-3} + 4*10^{n-4} + 4*10^{n-5}}{6} = \frac{4*10^{n-5} ( 1+10+100+1000+10000 +10^5)}{6} =\frac{4*10^{n-5} *111111}{6}  Integer
SUFFICIENT

2)  n=10 
Since if we take n=10 and m=6 
The expression will not be divisible by 7
NOT SUFFICIENT

IMO A

Abhigmat2019 · Answer

Hi.....

Not convinced with the OA..... Let us consider n=10, m=3, which gives Avg as an Integer. However, if we consider n=10, m=5, its Avg will not be an integer. I had opted for Optn E.... Please clarify.

Bunuel   chetan2u  request your help

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.

For positive integers n and m (n>m), is the average (arithmetic mean)

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 Data Sufficiency (DS) Questions