Find the remainder when 71234 is divided by 10

Question

Interesting Applications of Remainders – Practice question 1

Find the remainder when  7^{1234}  is divided by 10?

A.  1
 B.  3
 C.  4
 D.  7
 E.  9

To solve question 2:    Question 2

To read the article:      Interesting Applications of Remainders

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Chethan92 · Accepted Answer

Numbers 2, 3, 7 and 8 have a cyclicity of 4.
Numbers 4 and 9 have a cyclicity of 2.
Numbers 0, 1, 5 and 6 have a cyclicity of 1.

The remainder when 1234 is divided by 4 is 2.
hence  7^{1234} = 7^2

So the remainder when 49 is divided by 10 is 9.

E is the answer.

EgmatQuantExpert · Answer

Solution

To find:  
 • The remainder, when  7^{1234}  is divided by 10

Approach and Working:  
 We know that if we divide a number by 10, then the remainder is same as the units digit of the given number.
 • Hence, we need to determine the units digit of  7^{1234} .

As per our conceptual understanding, the units digit cycle of 7 is as follows:
 •  7^1 = 7^5 = 7^{4k+1} = 7 
•  7^2 = 7^6 = 7^{4k+2} = 9 
•  7^3 = 7^7 = 7^{4k+3} = 3 
•  7^4 = 7^8 = 7^{4k+4} = 1

As we can write  7^{1234}  as  7^{4k+2} , the unit digit of the number is 9.

Hence, the correct answer is option E.

Answer: E

Abhishek009 · Answer

\frac{7^4}{10} = \frac{2401}{10} =  Rem  1

\frac{7^{1234}}{10} = \frac{7^{308*4+2}}{10}

\frac{7^{308*4}}{10}   will have remainder  1  &  \frac{7^2}{10}  will have remainder  9 , Thus Answer must be (E)

BrushMyQuant · Answer

What is the remainder when  7^{1234}  is divided by 10

Theory: Remainder of a number by 10 is same as the unit's digit of the number

( Watch this Video  to Learn  How to find Remainders of Numbers by 10 )

Using Above theory Remainder of  7^{1234}  by 10 = unit's digit of  7^{1234}

Now to find the unit's digit of  7^{1234}  , we need to find 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.
=> We need to divided 1234 by 4 and check what is the remainder
=> 1234 divided by 4 gives 2 remainder

=>  7^{1234}  will have the same unit's digit as  7^2  = 9
=> Unit's digits of  7^{1234}  = 9

So,  Answer will be E 
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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Find the remainder when 71234 is divided by 10

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