What is the remainder if 7^384 is divided by 5 ?

Question

What is the remainder if  7^{384}  is divided by 5 ?

A. 0
B. 1
C. 2
D. 3
E. 4

chetan2u · Answer

Remainder by 5 depends on units digit. 
The cyclicity of units digit of successive powers of 7 (  7^1, \ \ 7^2, \ \ 7^3.....  is 7, 9, 3, 1, 7, 9....
384 is a multiple of 4 as 84 is a multiple of 4.
So the units digit of  7^{384}  is same as that of  7^4 , that is 1.
So the remainder when 1 is divided by 5 is 1.

B

Abhishek009 · Answer

\frac{( 5 + 2)^{384}}{5}   = Rem  \frac{2^{384}}{5}

We know,  \frac{16}{5} =  Rem 1

So, we have  \frac{2^{3*128}}{5} , Since 284 is completely divisible by 3, Answer must be (B)

Villiavello · Answer

Learning this really helped me with reference to cyclicity:

When the numerator is has a lot of exponents, one divides the number by its cyclicity to find its own "personal" cyclicity.

Thus now,

7^1 / 5 = r2

7^2 / 5 = r4

7^3 / 5 = r3

7^4 / 5 = r1

----

Now we see that the new cycle has a cyclicity of 4, and the above remainders according to its cyclicity:

384 / 4 = r 0 (or the whole number, therefore it will be the last cycle):

r = 1.

(i.e. if the remainder was 1, the remainder would be the 1st cycle, being r2, etc.)

TestPrepUnlimited · Answer

7^4  ends in 1 so we can take away a power that is a multiple of 4 for free.  7^{384}  ->  7^{24}  ->  7^0 = 1 .

Thus the remainder is 1.

Ans: B

Kinshook · Answer

Asked: What is the remainder if  7^{384}  is divided by 5 ?

Remainder when 7^{384} is divided by 5 
= remainder when 49^{192} is divided by 5
= remainder when (-1)^{192} =1 is divided by 5 =1

IMO B

Posted from my mobile device

MathRevolution · Answer

Let's check the remainder of 7 raised to various power divided by 5.

\frac{7^1 }{ 5}  = remainder 2

\frac{7^2 }{ 5}  = remainder 4

\frac{7^3 }{ 5}  = remainder 3

\frac{7^4 }{ 5}  = remainder 1

=>  7^{384} = 7^{4 * 96} = 7^4

=>  \frac{7^4 }{ 5}  = remainder 1

Answer B

BrushMyQuant · Answer

We know to find what is the remainder when   7^ {384}   is divided by 5

Theory: Remainder of a number by 5 is same as the Remainder of the unit's digit of the number by 5

(  Watch this Video   to Learn How to find Remainders of Numbers by 5)

We can do this by finding the pattern / cycle of unit's digit of power of 7 and then generalizing it.

Unit's digit of  7^1  = 7
Unit's digit of  7^2  = 9
Unit's digit of  7^3  = 3
Unit's digit of  7^4  = 1
Unit's digit of  7^5  = 7

So, unit's digit of power of 7 repeats after every  4^{th}  number.
=> Remainder of 384 by 4 = 0
=> Units' digit of   7^ {384}   = Units' digit of  7^{Cycle}  = Units' digit of  7^{4}  = 1

=> Remainder = Remainder of 1 by 5 = 1

So,  Answer will be B 
Hope it helps!

MASTER How to Find Remainders with 2, 3, 5, 9, 10 and Binomial Theorem

https://www.youtube.com/watch?v=h1HTEhmUb1w

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What is the remainder if 7^384 is divided by 5 ?

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What is the remainder if 7^384 is divided by 5 ?

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