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.

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.
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:

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

Find the unit digit of 1234¹³²× 5247²⁵⁶× 1353¹³³?
Find the unit digit of 1234¹³²+ 5247²⁵⁶+ 1353¹³³?