What is the remainder when 7^11 + 7^111 + 7^1111 is divided by 8?

Question

What is the remainder when  7^{11} + 7^{111} + 7^{1111}  is divided by 8?

A. 0
B. 1
C. 3
D. 5
E. 7

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Aryaa03 · Accepted Answer

IMO-Option-D 
I tried it with a slightly different approach than what was already posted by a few other community members.
Feel free to point out errors and blunders, if any.

7^1 =7 ; division by 8, leaves a remainder of 7
7^2 = 49;division by 8, leaves a remainder of 1
7^3 = 343 ; division by 8, leaves a remainder of 7
7^4 = 2401 ; division by 8, leaves a remainder of 1 and so on.
As per this pattern,  odd powers of 7 leave a remainder of 7 and even powers leave a remainder of 1

Back to the question:
 7^{11}  leaves a remainder of 7
 7^{111}   leaves a remainder of 7
 7^{1111}  leaves a remainder of 7

Total Remainder = 7 + 7 +7 = 21
Adjusting extra remainder when divided by 8 
==> 21/8 leaves a remainder of 5 (2*8+5 = 21)

VivekSri · Answer

OA is D

PranavVyas · Answer

7^1=Remainder is 7 when divided by 8
7^2=Remainder is 1 when divided by 8
7^3=Remainder is -1 when divided by 8
7^4=Remainder is 1 when divided by 8
7^5=Remainder is 7 when divided by 8
7^6=Remainder is 1 when divided by 8
7^7=Remainder is -1 when divided by 8
7^8=Remainder is 1 when divided by 8
7^9=Remainder is 7 when divided by 8
⋮
​pattern will continue, for eg; 7^3 = 343/8 remainder will -1 , 7^3=-1

Now 7^11 + 7^111 + 7^1111 = -1+-1+-1

Therefore, 8-3 = 5 
IMO=D

unraveled · Answer

What is the remainder when  7^{11} + 7^{111} + 7^{1111}  is divided by 8?

A. 0
B. 1
C. 3
D. 5
E. 7
Whenever such a pattern is there some form of other pattern need to be identified to reach a solution.
Cyclicity of 7 may help here but considering the fact that it is raised for 11, 111 and 1111 times.
Hence what remainder is left after dividing 7(when raised to some power) by 8 needs to checked for atleast 4 times.

Here, for every odd number times when 7 is multiplied the remainder is 7. R \frac{7}{8}  = 7
For even number of times it is 1. R \frac{7^2}{8}  = 1

Now the powers 11, 111 and 1111 are odd so we have remainder as 7 thrice, totalling 21 which when divided by 8 leaves 5 as remainder.

Answer D.

versewriter · Answer

In remainders we have two rules
R(A+B) = R(A) + R(B)
R(A*B) = R(A) * R(B)

so, the equation will be
7^11/8 + 7^111/8 + 7^1111/8

As we already know that if 49 is divided by 8 will give you 01 as remainder,

so we wright the equation 7^11/8 + 7^111/8 + 7^1111/8 as -

= 7*1/8 + 7*1/ 8 + 7*1/8

= 21/8

So the remainder is 5

u2409 · Answer

7 can be rewritten as 8-1

when n is odd (a-1)^n takes form of (a^n+ak-1)

so all three parts the full form will take the following form (8^n+8k-3)/8
-3 is a congruent to 5 mod 8 so the remainder is 5.

Devil90 · Answer

We can use negative reminder theorem. 
 (-1)11 + (-1)111 + (-1)1111 =-3
Remainder= 8-3
 =5

#Smart_math

What is the remainder when 7^11 + 7^111 + 7^1111 is divided by 8?

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What is the remainder when 7^11 + 7^111 + 7^1111 is divided by 8?

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