Remainder theorem

Question

Find the remainder $\dfrac{(39)^{42}}{10}$

Comments

My answer is 1, but the actual answer is 9??

I found this question on some youtube channel

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Yes you are correct.. answer is 1 only

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Remainder of expression:39⁴²   / 10  = (-1)⁴² / 10 = 1 

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@akash.dinkar12 your method is very tricky right?? 

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No its simple method.

like when we divide 39 by 10 we get remainder 9 or -1, that is the reason I m raising -1 to power 42

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Negative remainder concept is used by him..it is also fine even gives the answer faster then any method in this particular question

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(-1)$^{42}$ = 1 right??

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yes Lakhsman

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Thanks

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$\rightarrow$ $(39)^{42}$ mod $10$

$\rightarrow$ $(1521)^{21}$ mod $10$

$\rightarrow$ $(1)^{21}$ mod $10$

$\rightarrow$ $1$ mod $10$

$\rightarrow$ $1.$

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Yes Ans is 1

Answer 1

Calculate Euler Totient number of $10=2*5$ i.e $\phi(10)= 10*(1-\frac{1}{2})*(1-\frac{1}{5})$

$\phi(10)= 4$

Using remainder theorem:

$\frac{rem(\frac{39}{10})^{rem({\frac{42}{4}})}}{10}$

$\frac{{9}^{2}}{10}= \frac{81}{10}=1(rem)$

Hence 1 is the correct answer

https://www.mbatious.com/topic/61/remainder-theorem

Comments

@ srestha 

you see akash.dinkar12 comment nicely explain

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yes
got this link https://www.mbatious.com/topic/61/remainder-theorem
I wanted to know if it is generalized formula or not

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@srestha

Yes this is genreralized Euler Remainder theorem.

Answer 2

You are dividing a number by 10.This means if the number is divisible 10 then it will leave remainder 0 and if not then it will leave the last digit as remainder.

The unit's place of 39 is 9. 39⁴² means you are multiplying 39 for 42 times.

39 x 39 = y ,where the unit place of y is 1 because 9x9 =81 and unit place of 81 = 1

39 x 39 x 39 = 39 x y = z ,where the unit place of z is 9. Previously y = 1.Multiply 1 x 9 is same as 9.

so

39^a % 10 = 1 when a is even

39^a % 10 = 9 when a is odd

Here a = 42 =even

Hence remainder will be 1

ANSWER = 1

Answer 3

(39⁴²)/10 in this problem reminder is 1.

You think this way --> 39*39*39......39

 now  the reminder term depends upon the least significant digit because of denominator 10.

So for 9*9*9*9.......9 now for 9ⁱ where i=2n+1 the least significant digit is 9 so reminder is 9 and for all 9ⁱ where i=2n least dignificant digit is 1 so reminder is 1.

So in our question 9⁴²=9²ⁿ where n=21,

Ans is 1.