Question

show that 3^n*4^m cannot end with the digit 0 or 5 for any natural number

Answer 1

4- 2x2
4^m - 2^2^m
if it end with 0 or 5 it should be a factor of5
therefore it won't end with zero

Answer 2

Let us take example of a number ending with 0

10=2×5

100=2×2×5×5

Therefore, number ending with 0 has 2 and 5 as the only prime factors

Whereas, 3^n=(3×1)^n

It is not in form of 2^m×2^n

Therefore, 3^n cannot end with 0 or 5

Similarly,

4^n=(2×2)^n

5 is missing in prime factors of 4

Therefore, it cannot end with 0 or 5