Concept of Unit Digit

To find the unit digit in (xyz)ⁿ

Here xyz is some number, n is the Index and z is the base unit.

We need to find the unit digit of (xyz)ⁿ

Rule:
Divide the index ‘n’ by 4, the remainder (R) can be 0, 1, 2 or 3.

1. If R = 0, then
• If base unit is odd except 5, then the unit digit will be 1.
• If base unit is even, then the unit digit will be 6.
2. If R = 1, 2 or 3, then
   unit digit =  unit digit of (base unit)^R

Note: if base unit is 0, 1, 5 or 6, the unit digit will be same as base unit.

Problems:

1. Find the unit digit of (1237)¹³²132/4 leaves remainder 0.
   As base unit is odd, the unit digit of (1237)¹³² is 1.
2. Find the unit digit of (1238)¹³²132/4 leaves remainder 0.
   As base unit is even, the unit digit of (1238)¹³² is 6.

Exercise:

1. Find the unit digit of 1234¹³² × 5247²⁵⁶ × 1353¹³³?
2. Find the unit digit of 1234¹³² + 5247²⁵⁶ + 1353¹³³?