If N=(307)
38
+(524)
20
, then what is the unit digit of N?

To find the unit digit of N, we find the unit digit of each individual term using the rules of cyclicity.
First Term: (307)
38


The unit digit depends only on the base's unit digit, which is 7.

The cyclicity of 7 is 4. Let's divide the exponent 38 by 4 to find the remainder:

38÷4=9 with a remainder of 2
Therefore, the unit digit of (307)
38
is the same as the unit digit of 7
2
=49, which is 9.

Second Term: (524)
20


The unit digit depends only on the base's unit digit, which is 4.

The pattern for the unit digit of 4
n
is:

4
1
=4 (odd powers end in 4)

4
2
=16 (even powers end in 6)

Since the exponent 20 is an even number, the unit digit of (524)
20
is 6.

Unit digit of N:

Unit Digit=9+6=15→5