These questions usually call for a divide and conquer approach. That is, simplify the problem by breaking it into bits.
Note that , and that . Thus . Note that , and . So . Note that , so . So .a)Find the remainder when 5^10 is divided by 19.
To find the last digit of this expression, consider it modulo 10 (for example, the last digit of 718 is ). Let's start with the small factorials : , , . Now . Next one : . Continue like this up to 10! (note that since it divides 10) ,then sum up all the last digits and take the last digit of this sum and you are doneb)Find the final digit of 1! + 2! + 3! + ... + 10!
Does it make sense ?