What is the remainder when 2^{18} + 2^4 is divided by 2^7?

Question

What is the remainder when  2^{18} + 2^4  is divided by  2^7 ?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 16

agrasan · Answer

When we divide (2^18 + 2^4) by 2^7,

We get 2^11 + (2^4/2^7)

Simplifying further, 2^11 + (16/128)
Clearly, 16 is the remainder.

(E) is the answer.

SourNoun · Answer

18 > 7 so 2^18 is dividable to 2^7
4 < 7 so 2^4 isn't dividable to 2^7
So the remainder will be 2^4 = 16
Answer is E

vasu1104 · Answer

here 2^18 is divisible by 2^7 completely.

but 2^4 is not divisible by 2^1-7.
simplifying it, 16/128.

remainder is 16.

ans E

dotsoftme · Answer

when i am trying to solve this question, i am getting asswer as 1

when we add 2 power 18 + 2power 4 
and divide it by 2 power 7
common power cancel and answer remains of 16385 / 8 which give reminder 1

NoRest · Answer

Hey,

whenever you cancel a term, like you said power cancel 2^4 its a rule of division and remainders that you have to multiply it back when you get the remainder, for instance if you have to find remainder of 36/16,

1. You can see the remainder is 4 
2. But if you cancel both the terms by 4 you will get 9/4 and here the remainder is coming as 1
3. But since you cancelled both the terms by 4 you have to multiply the 4 back again to the remainder 1

Hope this helps.

chetan2u · Answer

Note: Remainders get added and multiplied..

Example ..  
 3+15 divided by 11 will give 3+(15-11) or 7 as remainder.  =>>18 divided by 11 also gives 7 as remainder.  
 3*15 divided by 11 will give 3*(15-11) or 12, that is 12-11 or 1 as remainder.   =>>45 divided by 11 also gives 1 as remainder.

Knowing the above, let us tackle the question at hand.

Remainder when   2^{18} + 2^4  is divided by  2^7  is same as remainder when (   2^{18}   divided by  2^7 ) + (  2^4  divided by  2^7 ).
Now   2^{18}   is multiple of   2^7  , so remainder will be 0, while   2^{4}   is less than   2^{7}   so   2^4   itself will be the remainder.
Hence answer is 0+  2^{4}   or 16.

E

egmat · Answer

Hi dotsoftme,

Your factoring is actually correct, and I like that you spotted the common power. The problem is in the very last step, when you read the remainder straight off the reduced fraction.

Here's what you did:
-  2^18 + 2^4 = 2^4 · (2^14 + 1) = 2^4 · 16385 
- Dividing by  2^7  and cancelling  2^4  top and bottom gives  16385 / 2^3 = 16385 / 8 
-  16385 = 2048 × 8 + 1 , so that reduced fraction leaves remainder  1

All true - but that  1  is not the answer to the original question.

The rule you missed

When you cancel a common factor  k  from both the number and the divisor, the remainder gets divided by  k  too - so you must  multiply it back  at the end.

Why? If a number splits as  a = (divisor)·q + r , then multiplying everything by  k  gives  k·a = (k·divisor)·q + k·r . The remainder scales by the same  k .

You cancelled  2^4 = 16 , so the real remainder is  1 × 16 = 16 . That matches choice  E , and it's exactly the  16  that agrasan and the others got by keeping  2^4  intact as  16/128 .

Quick check with small numbers

Find the remainder of  20 ÷ 8 : it's  4  (since  20 = 2×8 + 4 ).

Now cancel the common factor  4 : you get  5 ÷ 2 , remainder  1 . Multiply that  1  back by the  4  you cancelled -  4 .

So cancelling to simplify is fine - just remember to scale the remainder back up by whatever you cancelled.

Answer: E

What is the remainder when 2^{18} + 2^4 is divided by 2^7?

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What is the remainder when 2^{18} + 2^4 is divided by 2^7?

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