What is the remainder when 11^1 + 11^2 + 11^3 + ... + 11^11 is divided

Question

What is the remainder when 11^1 + 11^2 + 11^3 + ... + 11^11 is divided by 22 ?

A. 0
B. 1
C. 10
D. 11
E. 21

M40-04

Hint: this question has 30 sec solution. Do not overcomplicate it.

Bunuel · Accepted Answer

What is the remainder when  11^1 + 11^2 + 11^3 + ... + 11^{11}  is divided by 22 ?

A. 0
B. 1
C. 10
D. 11
E. 21

11^1 + 11^2 + 11^3 + ... + 11^{11}= 
 
 =11^1 + 11(11 + 11^2 + ... + 11^{10}) 
 
Observe that within the parentheses, we have the sum of 10 odd numbers, which results in an even number. Thus:
 
 11^1 + 11(11 + 11^2 + ... + 11^{10})= 
 
 =11^1 + 11*	ext{{even}}= 
 
 =11^1 + 	ext{{a multiple of 22}} 
 
Therefore, the remainder when the above expression is divided by 22 is 11.

Answer: D

gmatophobia · Answer

Rem( \frac{11(1 + 11 + 11^2 + ..... + 11^{10}) }{ 11 * 2} ) We can cancel 11 from numerator and denominator Rem( \frac{1 + 11 + 11^2 + ..... + 11^{10} }{ 2} ) 1 + 11 + 11^2 + ..... + 11^{10} are 11 terms ( 11^0 to 11^{10} ) and each term is odd, so the sum of 11 odd terms is odd. Remainder(odd / 2) = 1 As we cancelled 11 initially, net remainder = 1 * 11 = 11 Option D P.S. Not sure if this was the 30 second solution

Bunuel · Answer

That's close to what I had in mind and basically uses the same logic.

AritraKundu · Answer

Hi, here is my working but please let me know if my thought process is correct.

11^1+11^2+11^3+....+11^11/22 can be broken into
11^1/22, 11^2/22,11^3/22........11^11/22
In all cases 11 will be the remainder
So, 11+11+11+....+11/22=11*11/22=121/22
The remainder is 11

hardikpujara · Answer

not sure about 
As we cancelled 11 initially, net remainder = 1 * 11 = 11

gmatophobia · Answer

The general rule is - "Whenever we cancel a common factor from the numerator and denominator , that factor needs to be accounted for in the final remainder"

Example : Say we want to find the remainder of Reminder( \frac{22}{55} )

By logic we know that the remainder is 22.

Let's try to do this mathematically -

Remainder( \frac{22}{55} )

We can cancel 11 from both the numerator and the denominator and arrive at a reduced fraction  \frac{2}{5}

The remainder when 2 is divided by 5 is 2.

So is the remainder when  \frac{22}{55}  = 2 ?  No !

That's because we didn't factor the 11 that we cancelled out initially.

The net remainder = 2 * 11 = 22.

Hope it's clear !

khumoyuns · Answer

You may break this number down to:

(11 + 11*(11+11^2...+11^10))/22

Expression in the inner bracket should be even (as it is the sum of an even number of odd numbers);
11 * by even number should be divisible by 22;

You will end up with q (some integer) + 11/22. Thus, the remainder should be 11

Hope the above is helpful

sachi-in · Answer

What is the remainder when 11^1 + 11^2 + 11^3 + ... + 11^11 is divided by 22 ?

A. 0
B. 1
C. 10
D. 11
E. 21

11 + 11^2( 1 + 11 ) + 11^4( 1 + 11 ) + ..... + 11^10( 1 + 11 )

11 +   12 * ( 11^2 + ... 11^10)

Highlighted portion multiple of 22

remainder 11

Oppenheimer1945 · Answer

Odd terms hence Expression =11(odd no)=11(2k+1)=22k+11

Leaves remainder of 11 when divided by 22

Posted from my mobile device

Yuvrajbangatrader · Answer

11/22>> remainder is 11
121/22>> remainder is 11
Pattern "Might" follow till 11^11
Therefore 11+11+11+11+...=121
and 121/22>> remainder 11

sujoykrdatta · Answer

There are 11 terms. If we ignore the first term, we have 10 terms, all divisible by 11 and all odd
Thus, sum of the last 10 terms is divisible by 11 and also by 2 i.e. also by 22

Thus, the remainder is simply the first term, i.e. 11 when divided by 22

shubhim20 · Answer

thanks you explained it so well

Kinshook · Answer

What is the remainder when 11^1 + 11^2 + 11^3 + ... + 11^11 is divided by 22 ?

The remainder when 11^1 is divided by 22 = 11
The remainder when 11^2 is divided by 22 = 11
The remainder when 11^3 is divided by 22 = 11
...

The remainder when 11^11 is divided by 22 = 11

The remainder when 11^1 + 11^2 + 11^3 + ... + 11^11 is divided by 22
= The remainder when 11 + 11 + 11.... 11 times is divided by 22
= The remainder when 11*11 = 11^2 is divided by 22 = 11

IMO D

What is the remainder when 11^1 + 11^2 + 11^3 + ... + 11^11 is divided

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What is the remainder when 11^1 + 11^2 + 11^3 + ... + 11^11 is divided

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