Question

Show that $3^n\times 4^m$ cannot end with the digit 0 or 5 for any natural numbers ${}^{\prime }{n}^{\prime }$ and ${}^{\prime }{m}^{\prime }$

Answer 1

S.T $3^n\times 4^m$ cant end with $0$ or $5$.

For a number to end with $0$, the number

must have $2$ & $5$ as factors.

$10=2\times 5$

$100=2^2\times 5^2$

For a number to end with $5$, the number must have $5$ as a factor.

$30=5\times 6$

$3^n\times (2^2{)}^m=3^n\times 2^{2}m$

This number dosent have $5$ or $5$ and $2$ as its factors. Therefore the number cant end with $0$ o $5$.