Question

find the unit's digit in the product (2467)^153 * (341)^72

Answer 1

341 to any power will always have a 1 in the unit's position (1*1=1)

Answer 2

for 2467, concentrate on 7 to a power:
7^0 =0
7^1 = 7
7^2 = 49
7^3 = 343
7^4 = 2401
7^5 = 16807

looking at just the unit's digit
we see it repeats after 7^4

Answer 3

that means we should write
2467^153 as 2467^(4*38) * 2467^1
and then
\[ \left(2467^4\right)^{38} \cdot 2467 \]

Answer 4

as noted, the 2467^4 ends in 1, and any power of it ends in 1
we are left with a 7 from 2467 as the unit's digit of the product