Finding remainder

Question

What is the remainder when 473³⁸ is divided by 5?

Answer 1

numbers that are divisible by 5 ends either in a 5 or a 0. Examples are 5, 10, 15, 20, 25, and so on. The last digit of 473³⁸ can be found by finding the last digit of 3³⁸. 

3¹ is 3

3² is 9

3³ is 27

3⁴ is 81

3⁵ is 243

 

The last digits for the above series ends in 3, 9, 7, 1 and then 3. The sequence repeats after every four numbers. So divide the exponent by 4 and find the remainder. Here 38 divided by 4 leaves a remainder of 2. This means we have to look at the second number in the sequence which is 9.  which means 3², 3⁶, 3¹⁰, 3¹⁴, ... 3³⁸ all end in 9. So 3³⁸ also will end 9. 

 

The number that is divisible by 5 ends in a 0 or 5. The number 473³⁸ which ends in 9 when divided by 5 should give a remainder 4.