Question

remainder=?

5⁶²⁵/7=?

Comments

They asked for remainder or something else ?

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remainder===?

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Remainder=4

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Find the remainder $\frac{5^{625}}{7}=?$

$5^{1}=5$

$5^{2}=25$

$5^{3}=125$

So,we can say that cyclicity of $5$ is $1$ and $5^{n}=5$(Unit digit)$,n\epsilon$ Natural number.

So,$\frac{5}{7}$ gives remainder $5.$

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[Using Totient method]

-find Toient of 7 =7(1-1/7)=6

625 mod 6=1

$5^{1}$ mod 7=5

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@Lakshman Patel RJIT by your method remainder of 25/7 should be 5 but its 4.Thats not working

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See the best-selected answer.

Answer 1

$5^{625} = {5^{5}}^{125} = 3125^{125} $

$3125^{125} \mod 7 = (3125 \mod 7)^{125} \\= 3^{125} \mod 7 \\= \left(3^5 \mod 7\right) ^{25} \\= \left(243 \mod 7\right) ^{25} \\= 5^{25}  \mod 7 \\=\left(5^5 \mod 7\right)^5\\= \left(3125 \mod 7\right) ^ 5 \\= 3^5 \mod 7 = 5.$

Comments

Can also be done using fermet's little theorem :  7 is prime and 5 is prime to 7 .... so   (5^6)/7 has remainder 1 . Using the above relation   {(5^(6*104) ) * 5} /7 = (1 * 5) /7  ... remainder is 5.

Answer 2

5 mod 7 = 5 

25 mod 7 = 4

125 mod 7 = 6

625 mod 7 = 2 

3125 mod 7 = 3

15625 mod 7 = 1 

If we take number pattern repeats like this { 5 4  6 3  2  1.........} 

for 5^624 mod 7 =  1 so next number will be 5 which is 5^625