number properties

Hi,
Each number follows a certain pattern when it is raised to some power..
It is generally set of 4 unit digits in most cases, and in other less than 4..
so keeping this we can find the last digit ..

now to your Q where you have to subtract a bigger number from a smaller number..
find the numeric value without sign, that is subtract the numeric values..
lets see the Q
The last digit of 12^12 + 13^13 - (14^14 X 15^15) =

0
1
5
8
9

now (14^14 X 15^15) has to have a 0 as last digit..
12^12 will have 6 as last digit as any number with 2 as unit digit will follow 2,4,8,6,2,4,8,...
13^13 will have 6 as last digit as any number with 3 as unit digit will follow 3,9,7,1,3,9,7,1.....
so we have 6+3 being subtracted from 0, which is 1 ans B

Let's start with 12¹² + 13¹³. This will have the same units digit as 2¹² + 3¹³, so we can solve that instead.

Notice that the powers of 2 end 2, 4, 8, 6, 2, 4, 8, 6, ...

and that the powers of 3 end 3, 9, 7, 1, 3, 9, 7, 1, ...

So 2¹² + 3¹³ ends in 6 + 3, or 9.

Now we're subtracting (14^14 * 15^15). This is a really big number that ends in 0. Notice that 9 - 10 = -1, and that 9 - 20 = -11, and that 9 - 30 = -21, etc.

So something that ends in 9 minus something GREATER that ends in 0 will give us a negative that ends in 1. Success!