Question

Let n be a positive integer. Find the remainder when (5*31^12n) + (20*25^ (2n+1)) is divided by 13, and explain the steps.

Answer 1

Answer:

The answer is 11.

Step-by-step explanation:

Let's say we have  

31  = 5 (mod13)

312 = 25(mod13) = −1(mod13)

Therefore,

= (5×31^12n) + (20×25^2n+1)

=(5×(31^2^6n) + (20×25^2n+1)

= [(5×(−1)^6n) + (20×(−1)^2n+1)] (mod13)

= [(5×1) + (20×−1)] (mod13)  

= (5−20) (mod13) = −15 (mod13) = 11(mod13)

Remainder =11

Thus the remainder is 11.