1. What is the remainder when (53)^35 is divided by 13?
"Consider two positive integers a1 & b1, which leave a remainder r1 & r2 when divided by a number q. The product of the two numbers a1 & b1 when divided by q will leave the same remainder as the product of r1 & r2 when divided by q. For example, the numbers 23 and 13 when divided by the number 8 leave a remainder 7 and 5 respectively. 23x13 = 299 when divided by 8 leaves a remainder 3 7x5 = 35 also leaves a remainder 3 when divided by 3. Using the above logic and using the fact that 53 when divided by 13 will leave a remainder 1, we can say that 53^2 when divided by 13 will also leave a remainder 1. 53^3 will also leave the same remainder because 53^3 = 53 x 53^2. Similarly, 53^35 will also leave a remainder 1. For the more mathematically minded readers, we can arrive at the above result using Binomial Theorem. 53^35 = (52 + 1)^35 = 52^35 + (34 terms which all contain a power of 52) + 1 In the above expansion, since 52 is divisible by 13, all the terms are divisible by 13 except the last term which is the number 1."