Question

Show that 3^n ×4^m cannot end with the digit 0 or 5 for any natural number n and m

Answer 1

Answer:

Step-by-step explanation:

Because 3 has factors 1 & 3 & 4 has factors 2 power 2 ,So They cannot be ended with 0 or 5.

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Answer 2

Proof : Let a number whose factors are :

$\bold{\Longrightarrow{ 3^n \times 4^m}} \\ \\ \bold{ \Longrightarrow{x = 3^n \times 2^{2m}}}$

Here let the product of $\bold{ 3^n}$ and $\bold{2^{2m}}$ be x.

This means that x has no factor other than 3 and 2.

Thus, "5" can not be the factor of x or in more precise way 5 cant divide x.

Hence a number whose factor is not 5 can not end with 0 or 5.

( Reason : Like 100 have factor 2 × 2 × 5 × 5 or like 175 have 5 × 5 × 7. Hence its inportant for a number to have 5 as factor when ends up with 0 or 5. )

Q.E.D